Professor Nabil N Z Gindy
email: nabil.gindy@nottingham.ac.uk

Dr Maged S Morcos
email: maged.morcos@nottingham.ac.uk

Bülent Cerit

Allan Hodgson
email: allan.hodgson@nottingham.ac.uk

Gindy

M Cerit

Hodgson

Dr Bulent Cerit
email: bulent.cerit@nottingham.ac.uk

Mr Allan Hodgson

Strategic Technology Alignment Roadmapping STAR® Aligning R&D Investments with Business Needs

Keywords: Technology roadmapping, requirements capture, R&D, strategic technology alignment, portfolio balancing Technology roadmapping, requirements capture, R&D, strategic technology alignment, portfolio balancing

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Today's high technology manufacturing companies face an array of challenges arising from accelerating rates of technological change, globalisation of suppliers, competitors and markets and increasing customer expectations [START_REF] Conger | The pulse of industry[END_REF]. These challenges drive manufacturing companies to be more flexible and responsive in the integrated development and manufacture of innovative products; a more rapid and balanced response to the predictable and unpredictable changes of the manufacturing environment plays an important role in the long term success of companies (Gindy & Saad 1997).

The European Institute of Technology Management (EITM) defines technology management as the "…effective identification, selection, acquisition, development, exploitation and protection of technologies (product, process and infrastructure) which are needed to maintain a market position and business performance in accordance with the company's objectives". The acquisition of appropriate technologies to enable the development and manufacture of innovative products is of primary importance in enabling a competitive advantage to be gained [START_REF] Bucher | Integrated Technology Roadmapping: Design and Implementation for Technology-Based multinational Enterprises[END_REF]Narayanan & Jayaraman 1997;[START_REF] Farrukh | Industrial practice in technology planning -Implications for a useful tool catalogue for technology management[END_REF].

Scott (Scott, 2001) carried out an international survey to investigate the problems that companies face in the current business environment. Ten key problems were identified, of which five related directly to technology issues. 'Strategic planning for technology products' was identified as the most important of these ten key issues, followed by 'new product project selection'. Technology roadmapping is gaining increased recognition in industry as a flexible technology management tool for supporting businesses and improving their technology management processes through activities such as technology forecasting and technology planning [START_REF] Bucher | Integrated Technology Roadmapping: Design and Implementation for Technology-Based multinational Enterprises[END_REF]Farrukh, Phaal & Probert 2002;Grossman 2004;Phaal, Farrukh & Probert 2004a).

At the enterprise level, technology roadmapping is primarily a management tool to improve the enterprise's strategic technology planning processes by aligning technology acquisition to company strategic objectives derived from market and business drivers. In addition, the team-orientated technology roadmapping process also supports consensus building.

Background and state-of-the-art

Technology roadmapping began as an approach developed and applied by industrial practitioners, and more recently became a focus of academic research.

Technology roadmapping is now used at all levels in the global economy and can be used to co-ordinate science and technology developments at the international region or country level (e.g. European Union, USA, Canada, Japan), at the industrial sector level (e.g. semiconductor electronics), in multinational companies or at single-site enterprises.

Examples of roadmapping at the country level include the UK Foresight activities (Miles 2005), Canada (Industry Canada 2000), USA [START_REF] Wagner | Identifying critical technologies in the United States: a review of the federal effort[END_REF]. At the regional level, the European Union has developed roadmaps, based on the European Technology Platforms, as the basis for planning and encouraging ( [START_REF] Etn-See | Vision 2020 -the EMC technology roadmap, European Technology Network -Sustainable Electromagnetic Environments[END_REF] for examples of such roadmaps. At the industrial level, the most successful example of a long term roadmapping activity is the International Technology Roadmap for Semiconductors [START_REF] Sia | International Technology Roadmap for Semiconductors[END_REF]. Scenario planning [START_REF] Ringland | Scenario Planning -Managing for the Future[END_REF]) is a useful adjunct to national, international and industrial level roadmapping, as it encourages practitioners to consider not only the most likely futures, but also extreme futures. For example, the automotive industry should be aware of potential futures with major hydrocarbon shortages, or with international legislation that drastically reduces allowable carbon dioxide emissions.

The above roadmaps do not take into account the strengths and weaknesses of an individual enterprise, and it is at the enterprise level that we are seeing the most rapid developments in roadmapping. The earliest exponent, and still one of the leading users of technology roadmapping at the enterprise and supply chain level is Motorola. Faced with increasing product complexity and technical change rate, Motorola introduced its original paper-based roadmaps more than two decades ago [START_REF] Kappel | Technology Roadmapping: An Evaluation[END_REF][START_REF] Willyard | Motorola's Technology Roadmap Process[END_REF]. These paper-based roadmaps were later produced electronically, and can now be accessed and updated on-line. An introduction to Motorola's technology roadmapping approach can be found in [START_REF] Richey | Evolution of Roadmapping at Motorola[END_REF]. [START_REF] Phaal | Collaborative technology roadmapping: network development and research prioritization[END_REF], which has been applied in a wide range of companies. Table I presents a selection of technology roadmapping methodologies and their scopes.

Insert Table 1 here

Companies apply technology roadmapping primarily to ascertain what an evolutionary future is likely to offer, and to indicate actions that could be taken to be successful in such a future. It is carried out when organisations (departments, companies, sectors, countries or even economic regions) come to a decision point, for example prior to an annual allocation of funds or when a crisis due to a rapid loss of market share. • Second generation (mid-1980s to end-1990s): Methodologies aimed at improving strategic technology planning decisions.

• Third generation (end-1990s -today): Methodologies aimed at producing integrated technology management activities.

Third generation technology roadmapping is still largely at a developmental stage, and there is little supporting software to enable the integration of it into enterprise business processes. The STA Subgroup developed a first generation technology roadmapping methodology and applied it at a high technology industrial collaborator (Gindy et al, 2006). Steps of this methodology were also utilised for the purpose of requirements capture at several other companies and at industrial forums. The methodology enabled the assessment of R&D project proposals on the basis of four factors -investment and benefit (the conventional financial methods), opportunities and risks. Opportunities covered a wide range of factors including technology alignment to product and capability requirements, opportunities for partnerships and knowledge benefits.

Following feedback from several applications of the methodology, it was considered that it was difficult to determine the degree of (or adequacy of) the proposal's technology alignment to company product and capability requirements.

It was therefore decided to develop an enhanced methodology by adding a specific technology alignment factor that deals only with the alignment of the project proposal technologies to product and capability requirements now and in the future; this issue is detailed in [Gindy et al 2006], Section 5: Further work.

The enhanced methodology, referred to as the Strategic Technology Alignment Roadmapping (STAR

®

) system, has three major factors, used to evaluate R&D project proposals, each with two or three sub-factors, as listed below:

1. Economics (E):

• Financial investment • Product requirements capture

• Financial benefit
• Competitiveness -benchmarked capabilities

• Familiarity -technology watch 3. Synergy (S):

• Opportunities

• Risks

The addition of a specific technology alignment factor at the top level enables us to evaluate projects and to balance a portfolio taking account of the company's technology requirements. The relative importance of each of the top level factors (financial, technology alignment and synergy) can be decided directly by assigning weights to each.

STAR

® is described in detail in the following subsections.

Introduction to STAR ®

STAR® is a directed, "needs-driven", technology planning process aimed at aligning enterprise R&D activities to its business environment through identifying, selecting, prioritising and developing technologies to satisfy market and product needs, enterprise drivers and technology competitiveness position.

The process acts as an integrative framework for bringing together expert knowledge to collect, generate, organise and analyse technology planning information The tools used to assist team members in their decision making, in particular, the analytic hierarchy process (AHP), provide transparency in the process and build consensus and confidence in the technology priorities that are output from the process. Figure 2, shows the STAR ® framework with its three phases and illustrates the flow of activity from beginning to end.

Insert Figure 2 here

3.2 STAR ® processes
The STAR ® methodology is based on closely coupling several techniques and methodologies to provide an integrated framework for guiding and justifying investments in R&D projects to achieve the optimum project portfolio that supports enterprise business drivers. It is intended to be a third generation system, i.e. one that is aimed at supporting several technology management activities.

The amount of data and information generated during the roadmapping process is large, very diverse and potentially very valuable for many applications. This information must be collected and stored systematically to facilitate processing and decision-making, not just for a single roadmapping session, but for reevaluation and re-use within and externally to the roadmapping process.

Most of the data required for STAR ® is captured via in-house developed software tools and stored in a common knowledgebase, which covers business drivers, market drivers, market segments, products, technologies, etc., and their relationships and hierarchies. This integrative, interactive knowledgebase has been designed in generic form and can, without modification, handle any

STAR ® phases and components

Implementation of the STAR ® process can be broken down into three major phases as shown in Figure 3. These are described in the following subsections.

Insert Figure 3 here

Phase 1: Preliminary scoping and development of hierarchies

The main aim of this phase is to define the overall scope of the STAR ® process for the user's enterprise, to adapt the generic framework provided by STAR ® , collect and store the information that is required to support decision-making for technology acquisition projects, and to define the factors (and their weightings) used to assess technology project proposals.

The preliminary phase starts with the formation of a STAR ® team, consisting of personnel with appropriate levels of knowledge regarding the enterprise organisational structure, markets, current and future products, product life cycles, market and business drivers, supply chain, make and buy strategies, and the manufacturing technologies used in product manufacture. In order to be effective (in terms of knowledge and stakeholder commitment), the team will typically consist of personnel from a range of departments and at various levels in the organisational hierarchy.

The team activities in this phase include:

1. Development of a map of potential enterprise goals and business drivers, their definitions and linkages (e.g. cost reduction, quality improvement, delivery performance, agility and responsiveness). There may be several requirements capture hierarchies, each associated with a major product group. The requirements capture hierarchy is required in order to associate technologies (product, material, process)

with products and business drivers (see Figure 5). The hierarchical structure of a simple product may be very shallow; for example in the case of an electric kettle, it may be sufficient to consider only the product and first level assemblies and components. However, the hierarchical structure of a complex, technically-advanced product such as an aircraft may be very deep, and it may be necessary to consider product, assembly, sub-assembly, component, part family and features.

Insert Figure 5 methodology can be initially used even with a limited set of information, and more detail can be added as the data becomes available At the completion of the preliminary phase, the enterprise will have populated the STAR ® framework with the business, organisational, product and technology information that define the key areas of importance and interest to that enterprise.

It will also have identified and defined the various factors (financial, internal & external constraints, desirable and undesirable intangibles) that should be considered when evaluating R&D project proposals, for example collaboration, partnerships, subcontracting, funding, expertise, resources, intellectual property rights and confidentiality.

The information that is required is split into three major factors (as stated earlier)economics, technology and synergy. These enable proposed projects to be evaluated to express the alignment to these drivers:

Economic factors: Including financial investment and benefits.

Phase 2: Technology requirements

The main purpose of this phase is to evaluate and rank the technologies that are key to current and future enterprise success. This technology information provides the basis for creating R&D projects that are linked to business needs, and contributes to the assessment of R&D projects.

It is very important to achieve consensus in this phase, as the outputs of the phase define the technology priorities of the enterprise. A team-based approach is utilised (in particular in requirements capture, see Phase 2, Step 1), in which a facilitator encourages team members to discuss and come to a decision on the relative priority of each factor under consideration (e.g. technology).

This phase includes three steps:

Requirements capture: Aimed at information gathering to define the enterprise's key business drivers and the associated product and technology priorities.

Benchmarking: Aimed at assessing the enterprise competitive position in a range of technologies of high relevance to enterprise activities.

Technology watch: Aimed at assessing the state-of-the-art and forecasting technology progress in emerging technology areas of potential relevance to ongoing and planned future enterprise activities.

Based on this assessment, technology areas requiring further investigation can be highlighted.

The detailed activities of this phase are described below.

Phase 2 Step 1: Requirements capture

The STAR ® requirements capture hierarchy (see example in Figure 4) is used to rank and score products (to provide product priorities) and technologies (to The workshops originally utilised a proprietary analytic hierarchy process (AHP) package to rank the relative importance of business drivers, products and technologies; however, this was found to be unsatisfactory as, on occasions, workshop delegates later questioned the outcomes of the workshops. Therefore, an AHP-based software tool was developed to enable multiple layers in a requirements capture hierarchy to be ranked in a single session, to capture the key reasoning of the workshop team, to generate a report (confirming the activities and results) and to capture the results in the STAR © database. This enabled immediate printing and emailing of the whole session to delegates, and also enabled later workshops to re-open and continue the session, if required.

The requirements capture step provides inputs to the project assessment phase in the form of the product priority and technology priority factors.

Phase 2 Step 2: Benchmarking

The benchmarking step of the STAR ® process enables the enterprise to make an accurate assessment of its competitive position with regard to technology maturity (base, key, pacing and emerging). The aim is to ensure that the enterprise is ahead of the competition in technologies that provide its products and/or services with a competitive advantage, and that it is satisfactorily placed to exploit other technologies. For example, it may be unsatisfactory to be only two years ahead of the competitors in certain casting technologies but, at the same time, satisfactory to be one year lagging in certain machining technologies. includes the gathering of information on R&D, patents, applications, suppliers, competitor interest, displaced and alternative technologies, etc., from a range of internal and external sources. The output from this process is a standard structured 'emerging technology' report on each technology and a summary report that covers all of the selected emerging technologies.

The assessment of emerging technologies by a group of experts:

Following receipt of the emerging technology reports, the STAR ® AHP software tool (described earlier) can be used to assess the importance of each emerging technology. Alternatively, another approach, e.g. Delphi, or questionnaires (sent to internal experts and, potentially, technology suppliers and academics) can be used. In addition to the potential threats from, or benefits of, the technologies, it is also necessary to assess the likelihood of the technology reaching maturity, and the likelihood of the enterprise failing to track its process. To this end, a technological threats and opportunity analysis (TTOA) scoring system has been developed, akin to the wellestablished failure mode and effects analysis (FMEA) method.

The technology watch step provides an input to the project assessment phase in the form of a technology threat/opportunity factor for each technology.

Project creation and assessment phase

This is aimed at the generation of R&D projects proposals, and the selection of a balanced set of projects from these that together represent an optimised R&D portfolio aligned to enterprise requirements. This includes:

Project generation: Aimed at the generation of candidate R&D project proposals that address enterprise requirements described in (2) above.

Project assessment: Aimed at assessing the impact of R&D projects on enterprise objectives.

Portfolio balancing: Aimed at optimising the selection a group of R&D projects to maximize the impact of the projects on company operations within budget constraints.

Phase 3 Step 1: Adjustment of assessment settings

Aligning the project assessment settings (prior to the creation of Adjusting the project assessment settings: to ensure that projects that are aligned to the enterprise's technology requirements score more highly than those that are not. A statement indicating these general preferences is released to project proposal authors.

Phase 3 Step 2: Project generation

Information from phases 1, 2 and phase 3 step 1 are made available to the individuals or teams that are responsible for the development of project proposals.

In addition, company policies with regard to the current R&D round will be made clear, for example that there should be a certain percentage (by total cost) of projects aimed at technology insertion in ten to fifteen years time.

Although there is a risk that project proposals will reflect the individual biases of their developers, they will be aware that the proposals must pass predefined thresholds; this will encourage a strong measure of objectivity.

All project proposals must include in their descriptions sufficient information to allow the calculation of seven assessment factors used to evaluate each project:

• Project financial figures, financial investment factor (I) and expected project financial benefit factor (B);

• Project contributions to technology development (T), these include:

o Contribution towards the enterprise's product requirements placed on this technology. o Contribution based on the technology developments that may be profitable for the enterprise to exploit, or dangerous to ignore.

• Project synergy:

o Factors which enhance the chances of project success and/or produce further benefits to the enterprise (e.g. working with suppliers who will be responsible for implementing the technology, partnerships with leading edge research organisations); this enables the calculation of an opportunity factor (O).

o Factors which are considered as risks that may reduce the project's likelihood of delivering its expected non-financial outcomes or reduce the benefits that the company may get from the project, (e.g. collaboration with competitors, use of scarce resources); this enables a value to put on a risk factor (R).

Phase 3 steps 1 and 2 should result in the desired technology alignment of the submitted proposals.

Phase 3 Step 3: Project assessment

All project proposals can be assessed en masse, each with appropriate weighting given to its three main assessment factors (Economics =E, Technology=T and Synergy=S). Weightings can be used, for example, to favour certain types of projects (e.g. near term projects if it is desirable). The assessors (who will typically include some of the proposal developers or champions) utilise the information from phases 1 and 2 to assess the project proposals. Projects that fail to address any of the key technology issues are unlikely to be given approval.

Economic (E) assessment:

The economic assessment is based on the financial investment required by the project, and the financial return expected. Typically, this will be calculated using the enterprise's standard approach (e.g. net present value, discounted cash flow).

Technology (T) assessment:

Product priority (P): The project's contribution towards the enterprise's direct product requirements for the chosen technology is assessed from the project description. Combined with the product priority and technology priority factors, this enables the calculation of a technology priority factor (P).

Benchmarking -competitiveness (C):

Synergy (S) assessment:

The synergy assessment is based on an adapted list of opportunities and risks provided with the STAR ® knowledgebase, and produces values for project opportunities and risks.

A spreadsheet software tool is currently used to calculate project scores and to rank them (see Figure 7). It also provides information that may lead to rejection of individual proposals on the basis of thresholds that are associated with any of the assessment factors, for example lack of technology alignment, excessive risk, etc.

Phase 3 Step 3: Portfolio balancing

Project proposals should not be selected solely on the basis of their scores and ranks. It is important that an appropriately balanced portfolio is achieved in order to ensure that:

• an effective mix of technologies is developed for insertion into the enterprise,

• an appropriate balance is achieved between short-term and long-term projects, and

• utilisation of human, plant and other resources is within enterprise capacity.

Ultimately, the task of ensuring an effective portfolio will require the experience and judgement of a manager. Several graphical tools are provided in STAR ® to assist in this activity, as follows:

1-Technology frontier curves or graphs

The technology frontier is used in the second phase of STAR ® as a project visualisation technique; project proposals that appear on the frontier are not 'dominated' by other proposals in terms of their sets of attributes (in our caseeconomics, technology and synergy, or attractiveness and investment). The frontier analysis is useful in offering managers and decision-makers more flexibility to trade off any attractiveness dimension with any other dimension. For example, managers can choose to trade-off technology and synergy against pure monetary outcome.

As can be seen in Figure 8, a set of projects is displayed, showing those that are most efficient on the technology frontier curve (projects P16, P15 and P12). A trade-off between those efficient projects can then be made to favour one over the others. However, the process can also be enhanced by adding threshold limits for investment as a budget limits and/or attractiveness limits accepted by the group of decision makers, as in (Kirby & Mavris 2002) and as in Figure 8. Managers can then make a list of efficient projects resulting from the analysis and decide how much investment budget is required for implementation.

A detailed presentation on the application of the technology frontier to portfolio assessment (as part of a technology roadmapping exercise) can be found in (Cerit, Morcos & Gindy 2005). An explanation of the more general efficient frontier can be found in [START_REF] Markowitz | Portfolio Selection[END_REF]) and (Morcos & Singh 1992). The

2-Project cost/timeline scatter diagram

The project cost/timeline scatter diagram is a chart that presents the distribution of projects in terms of their costs against target implementation dates (e.g.

incorporation of the project technology into a product or process). It is produced by plotting (from Figure 7) project cost against technology target implementation date for each project, and provides a very useful view of the balance between near-term and long-term investment in technology. Figure 9 illustrates a project cost/timeline scatter diagram. This chart shows 55 projects, including the 15 projects (P11 to P25) shown in Figure 7.

3-Radar diagrams

Radar diagramming is a mapping technique in which axes representing key factors radiate out from a central point; the actual values for each project are then joined together, giving the appearance of a spider's web. Figure 10 shows a generic radar diagram with a scale from zero to ten on each axis. This allows the mapping of a set of project proposals on each individual axis or element to produce a set of nested or overlapping polygons.

A sample application

In this example, a set of 15 projects has been elicited in Phase 3, Step 1. These projects were actually generated within the research group, but were based in some cases on actual industrial projects, and in other cases on discussions with our industrial partners.

Industrial applications

STAR

® is being developed into an integrated system that captures, retains and shares information for re-use within STAR ® and by other functions. However, its full implementation represents a significant undertaking for an enterprise, and therefore its software-supported elements are also intended to be capable of application as independent tools. It is in this mode that STAR ® has been applied in industry to-date.

The requirements capture tool of STAR ® has been used and evaluated by our key collaborator, and has been adapted to meet its needs relating to the capture of informal discussion. The STAR ® technology watch methodology is being applied by the same collaborator in order to produce partially populated ontologies for two areas of technology; one of these areas currently imposes significant manufacturing constraints, the other area includes interesting emerging technologies. At the industrial sector level, the requirements capture and technology watch methodologies have been adapted and utilised to analyse process technologies of interest to the UK aerospace sector, and to produce recommendations for research investment in the medium-to-long term (Gindy, Hodgson and Johnston 2006).

The technology watch methodology has also been used at roadmapping workshops held for UK small/medium manufacturing enterprises (SMEs) to identify technology threats and opportunities arising from foreign competition and new materials and process technologies.

Future work

The STAR ® methodology is now sufficiently developed that it requires detailed evaluation of the integrated methodology, rather than evaluation of independentlyapplied elements, as has taken place to-date. . This will take account of prior commitments, and will result in improved realism in technology planning.

Conclusions

There is a clearly identified need for technology roadmapping, at the national, industrial and individual enterprise levels, in order to ensure the

  term research and development; see (I*PROMS 2006) and

Figure 1 Insert Figure 1 here•

 11 Figure 1 illustrates a simple company-level technology roadmap in the form of a

  technology hierarchy, drivers (e.g. strategic and local business drivers, market drivers, and 'voice of customer' drivers). It can also supply information for benchmarking, and partially populate the ontologies used by the technology watch functions (see later). It also captures the intermediate decisions and outputs of individual technology roadmapping activities, which can therefore be re-evaluated at a later stage.

2 .

 2 Adaptation of the generic structures and overall framework of STAR ® (markets, products, technologies and R&D projects etc.). 3. Mapping of the structure of the enterprise, its major functions and activities. 4. Mapping of the detailed descriptions and classifications (taxonomy) of the technologies of relevance to enterprise operations and any potential new and potentially disruptive or displacing technologies that may be useful to include in the data collection phase. Figure 4 illustrates the upper layers of such a taxonomy. Insert Figure 4 here 5. Development of the requirements capture hierarchy, to link enterprise business drivers to enterprise technologies. The requirements capture hierarchy should allow top-down decomposition of enterprise business drivers and the identification of the impact of the various technologies on product groups, and part families.

  Technology factors: Including technology requirements, technology benchmarking and technology forecast factors Synergy factors: Including opportunity and risk factors other than direct financial factors.

  ). This ranking activity is carried out via facilitated workshops, utilising an AHP tool.

1 . 2 .

 12 Figure6). The results are stored in the STAR ® database. The benchmarking step

  The identification of potentially competing emerging technologies: This is assisted by the above functional ontology sets, but is currently the least formalised part of the technology watch process.4. The application of the STAR® technology data collection process to each emerging technology: This process is a step-by-step approach that

  proposals): Based on the outputs from the requirements capture, benchmarking and technology watch activities (Phase 2, above), the technology manager fine-tunes the project assessment settings to reflect needs, e.g.: to score projects addressing current products above those addressing future products, to score projects addressing key technologies above those addressing base technologies, to score projects addressing high technology readiness levels (TRLs) above those addressing low target TRLs.

o

  Contribution towards the enterprise's required competitive position in the stated technology (e.g. 2 years ahead of the competition) based on the benchmarking requirements.

  The project's contribution towards the enterprise's required competitive position in the stated technology (e.g. 2 years ahead of the competition) is assessed, based on the benchmarking requirements (K2). This, combined with the technology priority and benchmarking level of concern factors, enables the calculation of a technology competitiveness factor (C) Technology watch -familiarity (F): The project's estimated contribution based on the technology developments that may be profitable for the enterprise to exploit (K3). This, combined with the technology priority and threat/opportunity factors enables the calculation of a technology familiarity factor (F).

  briefly below, and examples are provided later.

  also been used for investments decisions in technology management by(Kirby & Mavris 2000, 2001[START_REF] Duckles | s Center for Technology Roadmapping: A Resource for Research and Education in Technology Roadmapping[END_REF].

Figure 7

 7 Figure 7 illustrates an early STAR® project proposal assessment spreadsheet

Figure 1 :Figure 2 :Figure 3 :FigureFigure 8 :Figure 9 : 2 Figure 1 :Figure 2 Figure 7 :Figure 9 :Figure 10 :

 123892127910 Figure 1: Simple company-level technology roadmap

  To this end, members of the STAR

	Futher work includes the insertion of a full capacity-constrained, time-phased
	technology requirements planning capability into STAR
	F o
	r
	P
	e e r
	team are now discussing potential ways forward with the key industrial collaborator. R e v i e w	®
	Initial work with the collaborator will include tests of the STAR ® methodology using project proposal data from current or previous R&D funding rounds. The key O n
	outcome from this industrial evaluation is not intended to be the development of a l y
	company-specific version of the STAR ® methodology, but the enhancement of a
	range of adjustable parameters provided within the STAR ® methodology to enable
	it to represent a range of enterprise needs.
	29

®

  Morcos, M. S. and Singh, G., Portfolio analysis and its application to the various phases of civil engineering, in Information technology for civil and structural engineers", CIVIL-COMP, edited by Topping, B. H. V., and Khan, A. I., CIVIL-COMP press, 1992, Edinburgh. Narayanan, S. and Jayaraman, S., A knowledge-based decision support system for apparel enterprise evaluation, in Manufacturing Decision Support Systems, edited by Parsaei, H. R., S. Kolli, & Hanley, T. R., 1997, pp. 67-110. Phaal, R., Farrukh, C., and Probert, D., Structuring a systematic approach to technology management: concepts and practice, IAMOT, 2001, Lausanne. Phaal, R., Farrukh, C., and Probert, D., Fast-Start Technology Roadmapping, Centre for Technology Management, 2002, University of Cambridge.

	I*PROMS, Deliverable D5.13 -Updated roadmap covering all PAC research Michelle R. K. and Dimitri N. M., An Approach for the Intelligent Assessment of
	r r r r r implementation of effective technology acquisition programmes. F o Gindy, N. Z., Cerit, B., and Hodgson, A., Technology roadmapping for the next generation manufacturing enterprise, Journal of Manufacturing Technology Management, 2006, vol. 17, no. 4. areas, Project No. NMP2-CT-2004-500273, 2006, 61 pages. Industry Canada, Technology Roadmapping: A Strategy for Success, Industry Future Technology Portfolios, 40th AIAA Aerospace Sciences Meeting & Exhibit, 2002, Reno, NV Miles, I., UK Foresight: three cycles on a highway, International Journal of 6. F o F o Gindy, N. Z. and Saad, S. M., A Hierarchical model for responsive manufacturing, edited by. Gindy N. Z., Nottingham, UK, 1997,pp. 1-14. F o o F Foresight and Innovation Policy, 2005, vol. 2, no. 1, pp. 1-34.
	STAR ® is primarily intended for application at the enterprise level, where its software-based tools and shared database enable decisions to be revisited or re-evaluated with minimum effort. STAR P e e r P e e r P e e r P P Gindy, N.N.Z, Hodgson, A. and Johnston, S., Research Opportunities in Aerospace Manufacture -A NACAM White Paper. Report No. KTN/AeroDef Pub 0001/08/06, Issue 2, November 2006, 33 pages. e e r e e r
	To-date, the flexibility to allow users at the enterprise and industrial sector level to select and apply individual elements of the STAR ® methodology in order to obtain R e v i e w R e v i e w R e v i e w Grossman, D. S., Putting Technology on The Road, Research Technology Management, 2004, vol. 47, no. 2, p. 41. Grupp, H. and Linstone, H. A., National technology foresight activities around the R e v i e w R e v i e w
	quick answers to specific questions, has proved to be a significant asset. It is therefore important to ensure that technology roadmapping systems such as STAR O n O n O n globe: Resurrection and new paradigms, Technological Forecasting and Social Change, 1999, vol. 60, no. 1, pp. 85-94. O n O n ® can still offer 'quick and dirty' partial solutions to the problems of l y l y l y l y l y
	technology forecasting, selection and acquisition. Hodgson, A., Arman, H. and Gindy, N.N.Z., A technology watch function for the
	The STAR ® methodology is now at a stage of development that it requires a high technology enterprise, to be published in International Journal of Industrial
	detailed industrial evaluation of the full system. Discussions are in-progress with and Systems Engineering, 2007, vol.2.
	a view to such an evaluation.
	30 32 33 34

® represents an advance in the state of the art of enterprise technology roadmapping in that it offers technology managers the facility to tune the project assessment stage to reflect current priorities, i.e. to encourage the alignment of technologies to company requirements.

URL: http://mc.manuscriptcentral.com/tandf/tcim Email:ijcim@bath.ac.ukInternational Journal of Computer Integrated Manufacturing


Cédric Milliet
email: milliet@math.univ-lyon1.fr

On properties of (weakly) small groups

Keywords: Mathematics Subject Classification. 03C45, 03C60, 20E45, 20E99, 20F18, 20F24 Small group, weakly small group, Cantor-Bendixson rank, local chain condition, infinite abelian subgroup, group in a simple theory, infinite finite-by-abelian subgroup, nilpotent group

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ON PROPERTIES OF (WEAKLY) SMALL GROUPS C ÉDRIC MILLIET

Abstract. A group is small if it has countably many complete n-types over the empty set for each natural number n. More generally, a group G is weakly small if it has countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions for their traces in any finitely generated algebraic closure. An infinite weakly small group has an infinite abelian subgroup, which may not be definable. A small nilpotent group is the central product of a definable divisible group with a definable one of bounded exponent. In a group with simple theory, any set of pairwise commuting elements is contained in a definable finite-byabelian subgroup. First corollary : a weakly small group with simple theory has an infinite definable finite-by-abelian subgoup. Secondly, in a group with simple theory, a normal solvable group A of derived length n is contained in an A-definable almost solvable group of class n.

A connected group of Morley rank 1 is abelian [START_REF] Reineke | Minimale Gruppen[END_REF]Reineke]. Better, in an omegastable group, a definable connected group of minimal Morley rank is abelian. This implies that every infinite omega-stable group has a definable infinite abelian subgroup [START_REF] Cherlin | Groups of small Morley rank[END_REF]Cherlin]. Berline and Lascar generalised this result to superstable groups in [START_REF] Berline | Superstable groups[END_REF]. More recently, Poizat introduced d-minimal structures (englobing minimal ones) and structures with finite Cantor rank (including both d-minimal and finite Morley ranked structures). Poizat proved a d-minimal group to be abelian-by-finite [START_REF] Poizat | Quelques tentatives de définir une notion générale de groupes et de corps de dimension un et de déterminer leurs propriétés algébriques[END_REF]. He went further showing that an infinite group of finite Cantor rank has a definable abelian infinite subgroup [START_REF] Poizat | Groupes de petit rang de Cantor[END_REF]. More generally, we show in this paper that an infinite weakly small group has an infinite abelian subgroup, which may not be definable however. We then turn to weakly small groups with a simple theory. Recall that an ℵ 0categorical superstable group is abelian-by-finite [START_REF] Baur | Totally categorical groups and rings[END_REF]Baur,Cherlin and Macintyre]. In [START_REF] Frank | Small stable groups and generics[END_REF], Wagner showed any small stable infinite group to have a definable infinite abelian subgroup of the same cardinality. Later on, Evans and Wagner proved that an ℵ 0 -categorical supersimple group is finite-by-abelian-by-finite and has finite SUrank [START_REF] Evans | Supersimple ω-categorical groups and theories[END_REF]. We shall show that an infinite group the theory of which is small and simple has an infinite definable finite-by-abelian subgroup. However we still do not known whether a stable group must have an infinite abelian subgroup or not. Definition 1. A theory is small if it has countably many n-types without parameters for every natural number n. A structure is small if its theory is so.

Note that smallness is preserved by interpretation, and by adding finitely many parameters to the language. Small theories arise when one wishes to count the number of pairwise non-isomorphic countable models of a complete first order theory in a countable language. If such a theory has fewer than the maximal number of pairwise non-isomorphic models, it is indeed small. Note that ℵ 0 -categorical theories and omega-stable theories are small. Definition 2. (Belegradek) A structure is weakly small if it has countably many 1-types over a for any finite tuple a coming from the structure.

Weakly small structures were introduced by Belegradek to give a common generalisation of small and minimal structures. A weakly small ℵ 0 -saturated structure is small. Definition 3. (Poizat [START_REF] Poizat | Quelques tentatives de définir une notion générale de groupes et de corps de dimension un et de déterminer leurs propriétés algébriques[END_REF]) An infinite structure is d-minimal if any of its partitions has no more than d infinite definable subsets.

Provided that its language be countable, a d-minimal structure is weakly small as there are at most d non algebraic types over every finite parameter set, and fewer algebraic types than the countably many formulae. Note that weak smallness neither is a property of the theory, nor allows the use of compactness, nor guarantees that the set of 2-types be countable. It allows arguments using formulae in one free variable only. Those formulae, the parameters of which lie in a fixed finite set, are ranked by the Cantor rank and degree.

Examples. A non weakly small group. Let G be the sum over all prime numbers p of cyclic groups of order p. For every set of prime numbers P , the type saying that "x is p-divisible if and only if p is in P " is finitely consistent. This produces as many complete types as there are sets of primes, preventing G from being weakly small.

A non minimal, d-minimal group. Recall that a minimal group is abelian [START_REF] Reineke | Minimale Gruppen[END_REF]Reineke], and a d-minimal group is abelian-by-finite [START_REF] Poizat | Quelques tentatives de définir une notion générale de groupes et de corps de dimension un et de déterminer leurs propriétés algébriques[END_REF]Poizat]. Let M be a minimal group, and F a finite group of order d. Any semi-direct product M ⋊ F with a predicate interpreting M will do.

A non d-minimal, non small, weakly small group. Let p be a prime, and G the sum over all natural numbers n of the cyclic groups of order p n . The theory of G is the the theory of a Z-module, and eliminates quantifiers up to positive-prime formulae. So every definable subset of G is a boolean combination of cosets of subgroups of the form p n G, or p n x = 0. This allows only countably many 1-types over every finite subset, thus G is weakly small. But Fact 4.1 shows that it is not small.

The Cantor rank

Given a structure M , a set A of parameters lying inside M , and an A-definable subset X of M , we define the Cantor rank of X over A by the following induction :

CB A (X) ≥ 0 if X is not empty, CB A (X) ≥ α + 1 if
there are infinitely many disjoint A-definable subsets of X having Cantor rank over A at least α.

CB A (X) ≥ λ for a limit ordinal λ, if CB A (X) is at least α for every α less than λ.

If the structure is weakly small and if A is a finite set, this transfinite process eventually stops, and X has an ordinal Cantor rank over A.

The Cantor rank CB A (p) of a complete 1-type p in M over A is the least Cantor rank of the A-definable sets implied by p. It is also the derivation rank of p in the topological space S 1 (A) (sometimes plus 1, depending on the definition taken for the Cantor-Bendixson rank). The Cantor degree of X over A is the greatest natural number d such that there is a partition of X into d A-definable sets having maximal Cantor rank over A. We shall write dCB A (X) for this degree. It is also the number of complete types over A having maximal Cantor rank over A.

For a natural number n, we say that a map is n-to-one if it is surjective and if the cardinality of its fibres is bounded by n. Definable n-to-one maps preserve the Cantor rank, and the degree variations can be bounded by the maximal size of the finite fibres : Lemma 1.1. Let X and Y be A-definable sets, and f an A-definable map from X to Y . Then

(1) If f is onto, CB A (X) ≥ CB A (Y ). (2) If f has bounded fibres, CB A (Y ) ≥ CB A (X). (3) If f is n-to-one,
then X and Y have the same Cantor rank over A, and

dCB A (Y ) ≤ dCB A (X) ≤ n • dCB A (Y )
Remark 1.2. The first two points appear for one-to-one maps together with the introduction of Morley's rank [START_REF] Morley | Categoricity in power[END_REF]Theorem 2.3]. Poizat extends them for n-to-one maps in the context of groups with finite Cantor rank [18, Lemme 1] (independently to the author's work). To the author's knowledge, the result concerning the degree is new.

Proof. We may add A to the language. For point one, we show inductively that CB(X) is at least CB(Y ). If CB(Y ) ≥ α + 1, there are infinitely many disjoint definable sets Y 0 , Y 1 . . . in Y of rank at least α. Their pre-images are disjoint and have rank at least α by induction, so CB(X) ≥ α + 1. For point two, we show inductively that CB(Y ) is at least CB(X). Suppose CB(X) ≥ α + 1. In X, there are infinitely many disjoint definable sets X 0 , X 1 , . . . of rank at least α. As the fibres of f have cardinality at most n say, for every subset I of N of cardinality n + 1, the intersection i∈I f (X i ) is empty. Thus there is a subset J of N of maximal finite cardinal with 0 in J such that i∈J f (X i ) has the same rank as f (X 0 ). Put Y 0 = i∈J f (X i ). Iterating, one builds a sequence Y 0 , Y 1 , . . . of definable sets such that the sets Y i and f (X i ) have the same Cantor rank and CB(Y i ∩ Y j ) < CB(Y i ) for all natural numbers i = j. Inductively, one may cut off a small ranked subset from every Y i and assume that they are pairwise disjoint. By induction hypothesis, the rank of every Y i is at least α, so CB(Y ) ≥ α + 1.

For the third point, if Y has degree d, then there is a partition of Y in definable sets Y 1 , . . . , Y d with maximal rank. The pre-images of the sets Y i have maximal rank according to the first two points and form a partition of X, so the degree dCB(X) is at least dCB(Y ). For the converse inequality, let Y have degree d, and let Y 1 be a subset of Y of degree 1. It is enough to show that f -1 (Y 1 ) has degree at most n. Suppose there are n + 1 disjoint definable subsets X 0 , . . . , X n of f -1 (Y 1 ) with maximal rank. As the fibres of f have no more that n elements, the intersection n i=0 f (X i ) is empty, so there is a proper minimal subset I of {0, . . . , n} such that i∈I f (X i ) has the same rank as Y . Thus, the intersection of i∈I f (X i ) and f (X i ) has small rank for every i out of I, and dCB(Y 1 ) is at least two, a contradiction.

Remark 1.3. In Lemma 1.1(3), to deduce that X and Y have the same Cantor rank, the fibres of f must be bounded, and not only finite. Consider for instance Y to be the set of all natural numbers N together with the ordering, and X to be the set of pairs of natural numbers (x, y) so that y ≤ x. When projecting on the second coordinate, every fibre is infinite, so CB N (X) = 2 ; when projecting on the first coordinate, the fibres are finite, but still CB N (Y ) = 1.

Note that in the proof of Lemma 1.1, one can weaken the definability assumption on f , and simply assume that the image and pre-image by f of any definable set are definable. For instance, we easily get : Lemma 1.4. Let M be a model, X an A-definable subset of M , and σ any automorphism of the structure M . Then

CB A (X) = CB σ(A) (σ(X))
Definition 1.5. Let M be a structure, and X an acl(∅)-definable set in M . Let C be a monster model extending M . We consider the finite union of the conjugates of X(C) under the action of Aut(C). We write X for its intersection with M . Similarly, we define X to be the intersection of M with the finite intersection of the conjugates of X(C) under the action of Aut(C).

Note that neither X nor X depend on the choice of the monster model. X is a ∅-definable set containing X, whereas X is a ∅-definable subset of X.

If A is a subset of B and X an A-definable set, then CB A (X) is less than or equal to CB B (X). Note that the Cantor rank (respectively degree) of X over A or over the definable closure of A are the same. The Cantor rank over A also does not change when adding finitely many algebraic parameters to A, and the degree variation can be bounded : Lemma 1.6. Let X be a definable set without parameters, and let a be an algebraic element of degree n over the empty set. Then

(1) CB a (X) = CB ∅ (X) (2) dCB ∅ (X) ≤ dCB a (X) ≤ n! • dCB ∅ (X)
Proof. We assume in the proof that the language is countable. However, this assumption is not necessary (see Remark 1.9). For the first point, the Cantor rank of a set increases when one allows new calculation parameters, so CB a (X) is at least CB ∅ (X). Conversely, let us show that CB ∅ (X) is at least CB a (X). Suppose first that CB a (X) = ∞ holds. Then there must be 2 ℵ0 types over a in X. The restriction map from S(X, a) to S(X, ∅) is n!-to-one. Indeed, if x and y have the same type over ∅, there is a monster model C and an automorphism σ of C with y = σ(x). If q(x, a) is the type of x over a, then q(y, σ(a)) is the type of y over a. This shows that there are 2 ℵ0 types over ∅ as well, which yields CB ∅ (X) = ∞. So, we may assume that CB a (X) is an ordinal. Let us suppose that CB a (X) = α + 1 and that the result is proved for every ∅-definable set of CB a -rank α. There are infinitely many disjoint a-definable subsets X i of X, each of one having rank α over a. By Lemma 1.4 and induction hypothesis, for every i, the set X 1 and a conjugate of X i have the same rank (computed over the set ā of all conjugates of a). So a conjugate of X 1 intersects only finitely many X i in a set of maximal rank over ā. One can take off these X i , cut off a small ranked subset from the remaining X i and assume that the conjugates of X 1 do not intersect any X i . Iterating, one may assume that no conjugate of X i intersects X j when i differs from j. By Lemma 1.4 and induction hypothesis,

CB a (X i ) = CB ā(X i ) = CB ā(X i ) = CB ∅ (X i ) = α
As the sets X i are disjoint, CB ∅ (X) ≥ α + 1, so the first point is proved.

For the second point, we may assume that X has degree 1 over the empty set. Suppose that X has degree at least n! + 1 over a. Let X 1 be an a-definable subset of X with maximal rank over a and degree 1. The union X 1 of its conjugates has degree at most n! over a, so X 1 and its complement in X both have maximal rank over a, hence over the empty set, a contradiction. Definition 1.7. We shall call local Cantor rank of X over acl(a) its Cantor rank over any parameter b defining X and having the same algebraic closure as a.

Remark 1.8. In Lemma 1.6, if b is another algebraic parameter, one may have dCB a,b (X) > dCB a (X), so one need not have CB ∅ (X) = CB acl(∅) (X). In fact, CB acl(∅) (X) may not even be an ordinal. For instance, consider the unit circle S 1 = {x ∈ C : |x| = 1} with a ternary relation C(a, b, c) saying that b lies on the shortest path joining a to c. We add algebraic unary predicates A 1 , A 2 , . . . to the language, with A n = {x ∈ S 1 : x 2 n = 1} for every natural number n. This structure has CB ∅ -rank 0, but infinite CB acl(∅) -rank.

Remark 1.9. Lemmas 1.1, 1.4 and 1.6 are particular cases of a more general topological result. Let X be any Hausdorff topological space, X ′ his first Cantor derivative, and inductively on ordinals, let X α+1 stand for (X α ) ′ . The Cantor-Bendixson rank of X is the least ordinal β such that X β is empty and ∞ if there is no such β. Let us call a rough partition of X, any covering of X by open sets having maximal Cantor-Bendixson rank and small ranked pairwise intersections. The Cantor-Bendixson degree of X is the supremum cardinal dCB(X) of the rough partitions of X. Without compactness one could have dCB(X) ≥ ω. If X was a compact space, one could equivalently define CB(X) (which differs by 1 from the previous definition) by the following induction :

CB(X) ≥ 0 if X is not empty, CB(X) ≥ α + 1 if there are infinitely many open subsets O 1 , O 2 , . . . of X with CB(O i ) ≥ α and CB(O i ∩ O j ) < α for all i = j. CB(X) ≥ λ for a limit ordinal λ, if CB(X) ≥ α for every α < λ.
As an analogue of Lemma 1.1, replacing a "definable" set by an "open" set, and a "definable" map, by either a "continuous" map or an "open" one, we easily get : Lemma 1.10. Let X and Y be two Hausdorff topological spaces and let f be a map from X onto Y .

(1) If f is open and onto, then CB(X) ≥ CB(Y ). (2) If f is continuous and has finite fibres, then CB(Y ) ≥ CB(X). (3) If f is a continuous, open, n-to-one, then CB(X) = CB(Y ) and dCB(Y ) ≤ dCB(X) ≤ n • dCB(Y )
To deduce Lemma 1.1 from Lemma 1.10, we only need to pass from the category of definable sets to the category of topological spaces, and notice that an A-definable map f from X to Y induces a continuous open map f from the (compact) Hausdorff space of types S(X, A) to S(Y, A). Note that in Lemma 1.10(2), the map need only have finite fibres to get preservation of the rank, whereas it needs to have bounded fibres in Lemma 1.1 [START_REF] Baer | The subgroup of the Elements of Finite Order of an Abelian Group[END_REF]. Note also that f must have bounded fibres to ensure that f have finite ones. For Lemma 1.6, consider any continuous equivalence relation R on a Hausdorff topological space X, that is a relation such that the canonical map X → X/R is open. If every equivalence class of R has size at most some natural number n, as X/R is Hausdorff and as the map X → X/R is also continuous by definition, it follows from Lemma 1.10 that CB(X) = CB(X/R) and the inequalities dCB(X) ≤ dCB(X/R) ≤ n • dCB(X) hold. Let M be any first order structure, a an algebraic parameter of degree n, and C a monster model extending M . Applied to the space of types over a, modulo the equivalence relation "to be conjugated under the action of Aut(C)", the latter yields Lemma 1.6.

General facts about weakly small groups

As an immediate corollary of Lemma 1.1 we obtain a result of Wagner : Corollary 2.1. (Wagner [START_REF] Frank | Small stable groups and generics[END_REF]) If f is a definable group homomorphism of a weakly small group G, the kernel of which has at most n elements, then f (G) has index at most n in G.

Proof. Otherwise, one can find a finite tuple a over which at least n + 1 cosets of f (G) are definable, so G has degree over a at least (n + 1) • dCB a (f (G)), a contradiction with Lemma 1.1(3). Corollary 2.2. In a weakly small group, there are at most n conjugacy classes of elements the centraliser of which has order at most n.

Proof. Otherwise, let us pick n + 1 conjugacy classes C 1 , . . . , C n+1 of elements the centraliser of which has order at most n, and choose a finite tuple a over which these classes are definable. According to Lemma 1.1, each class C i has maximal Cantor rank over a and degree at least dCB a (G)/n, a contradiction.

For any set X definable in an omega-stable group, one can define the stabiliser of X up to some small Morley ranked set. In a weakly small group, we can define a local stabiliser up to some set of small local Cantor rank, where local means "in a finitely generated algebraic closure". We write A∆B for the symmetric difference of two sets A and B. Definition 2.3. Let X be a definable set without parameters in a weakly small group G, and let Γ stand for the algebraic closure of a finite tuple g in G. One defines the local almost stabiliser of X in Γ to be

Stab Γ (X) = {x ∈ Γ : CB x,g (xX∆X) < CB g (X)}
For any subroup δ of Γ, we shall write Stab δ (X) for Stab Γ (X) ∩ δ.

Corollary 2.4. Stab Γ (X) is a subgroup of Γ. If X is invariant by conjugation under elements of Γ, then Stab Γ (X) is normal in Γ.
Proof. Let a and b be in Stab Γ (X). The sets X, aX and bX have the same types of maximal rank computed over g, a, b, so CB g,a,b (aX∆bX) is smaller than CB g (X). As the rank is preserved under definable bijections, and when adding algebraic parameters, we have

CB g,a,b (aX∆bX) = CB g,a,b (b -1 aX∆X) = CB g,b -1 a (b -1 aX∆X) so b -1 a belongs to Stab Γ (X).
Recall that for a definable generic set X of an omega-stable group G, the stabiliser of X has finite index in G. For a weakly small group, we have a local version of this fact : Proposition 2.5. Let G be a weakly small group, g a finite tuple of G, and X a g-definable subset of X. If δ is a subgroup of dcl(g) and if X has maximal Cantor rank over g, then Stab δ (X) has finite index in δ.

Proof. Let m and l be the degree over g of G and X respectively. In G, there are m types of maximal rank over g which we call its generic types over g. Thus, for translates of X by elements of δ, there are at most C l m choices for their generic types. If one chooses C l m + 1 cosets of X, at least two of them will have the same generic types.

Weakly small groups definable over a finitely generated algebraic closure satisfy a local descending chain condition : Lemma 2.6. Let G be a weakly small group, and H 2 ≤ H 1 two subgroups of G definable without parameters.

(1) If H 2 ∩ acl(∅) is properly contained in H 1 ∩ acl(∅),
CB(H 2 ) = CB b (H 2 ) = CB b (bH 2 ) = CB b (bH 2 ) = CB(bH 2 )
It follows that CB(bH 2 ) is maximal in H 1 , so there must be only finitely many choices for bH 2 , and thus for bH 2 .

Theorem 2.7. In a weakly small group, the trace over acl(∅) of a descending chain of acl(∅)-definable subgroups becomes stationary after finitely many steps.

Proof. Let G 1 ≥ G 2 ≥ . . . be a descending chain of acl(∅)-definable subgroups.

According to Lemma 1.6(1), the local Cantor rank becomes constant after some index n. Then G i ∩ acl(∅) has finite index in G n ∩ acl(∅) for every i ≥ n after Lemma 2.6(2). Let a be some algebraic tuple such that G n is a-definable. By Lemma 1.6(1), we may add the parameter a in the language and assume without loss of generality that G n is ∅-definable. The intersection of the Gi ∩ acl(∅) when i ≥ n is the intersection of finitely many of them by Lemma 2.6(1) : it is a subgroup of G n ∩ acl(∅) of finite index, contained in G i for every i ≥ n. The sequence of indexes [G n ∩ acl(∅) : G i ∩ acl(∅)] is thus bounded, and bounds the length of the chain

G 1 ∩ acl(∅) ≥ G 2 ∩ acl(∅) ≥ . . . .
Remark 2.8. We shall call this result the weakly small chain condition. Note that Theorem 2.7 is trivial for an ℵ 0 -categorical group, and also if one replaces the algebraic closure by the definable closure.

The weakly small chain condition can be slightly generalised as follows.

Let X be a set, and E, F two equivalence relations on X. We write F ≤ E if F is finer than E, and F < E whenever F strictly refines E. If Y is a subset of X, the relation E may be restricted to Y . We write E ↾ Y this restriction. For every a in X, we write aE for the equivalence class of a, and denote the cardinal of the equivalence classes of E in X by |X/E|.

If F ≤ E, we call index of F in E (in X) the cardinal [E : F ] defined by sup{|aE/(F ↾ aE)| : a ∈ X}. Note that [E : F ] = 1 if and only if E = F . Moreover, if G is a third equivalence relation on X with G ≤ F ≤ E, then [E : G] ≤ [E : F ] • [F : G].
Theorem 2.9. In a weakly small structure X, let E 0 , E 1 , . . . be a descending chain of acl(∅)-definable equivalence relations on X, defined respectively over the algebraic tuples ē0 , ē1 , . . . . Assume that for all natural numbers i and all a, b in X, the classes aE i and bE i are in {a, b, ēi }-definable bijection. Assume also that for all i, a and b, the class a Ei and b Ei are in {a, b}-definable bijection. Then, the trace over acl(∅) of E i becomes constant after finitely many steps.

Proof. We begin by the analogue of Lemma 2.6.

Lemma 2.10. Let X be a weakly small structure, and E ≤ F two ∅-definable equivalence relation on X such that for every a, b in X, the classes aE and bE (respectively aF and bF ) are in {a, b}-definable bijection.

(1) If F ↾ acl(∅) < E ↾ acl(∅) , then for every a in acl(∅), either CB a (aF ) < CB a (aE), or dCB a (aF ) < dCB a (aE). ( 2) If for some a in acl(∅), the classes aE and aF have the same local Cantor rank, then the index

[E ↾ acl(∅) : F ↾ acl(∅)] is finite.
Proof of Lemma 2.10. As F ↾ acl(∅) < E ↾ acl(∅), one class e(E ↾ acl(∅)) must split into at least two disjoint classes e(F ↾ acl(∅)) and a(F ↾ acl(∅)). We may add this new constant e to the language and assume that eE and eF are ∅-definable.

The sets aF and eF are two ∅-definable disjoint subset of eE having same local Cantor rank, so the first point is proved. If aE and aF have the same local Cantor rank, then aF and aE do have the same rank over ∅, so there are only finitely many choices for aF , and thus for aF .

According to Lemma 1.1 and Lemma 2.10(1), the local Cantor rank of the E i -class of every algebraic element becomes constant after finitely many steps, so we may assume it is constant from E 0 . Then the index [E 0 ↾ acl(∅) : E i ↾ acl(∅)] is finite for every i after Lemma 2.10(2). Adding finitely many parameters to the language, we may assume that E 0 is ∅-definable. The conjunction of the Ei ↾ acl(∅) is the conjunction of finitely many by Lemma 2.10(1). It follows that the sequence of indexes [E 0 ↾ acl(∅) : E i ↾ acl(∅)] must be bounded. Proof. Note that the group has no second centre. Moding out the centre, we may suppose that the centre is trivial. If there is a non-trivial involution, every element is an involution and the group is abelian, a contradiction. Any non-trivial element g is conjugated to g -1 by some element, say h. So h is non-trivial and conjugated to h 2 , which equals h k for some k. Write δ for the definable closure of h and k. Since g is in C(h k ) and gh = hg, the element

h belongs to (C(C(h)) ∩ δ) \ (C(C(h k )) ∩ δ).
It follows that the chain

C(C(h)) ∩ δ > C(C(h k )) ∩ δ > C(C(h k 2 )) ∩ δ > • • •
is infinite, contradicting the weakly small chain condition 2.7.

Let G be any group. We say that a subgroup H of G is proper if it is not G.

Proposition 3.3. An infinite non-abelian weakly small group has proper centralisers of cardinality greater than n for each natural number n.

Proof. For a contradiction, let G be a weakly small counter example with all proper centralisers finite of bounded size n. Note that G has finite exponent, and a finite centre.

(1) The group G has finitely many conjugacy classes.

As the centralisers have bounded size, we apply Corollary 2.2. We may also add a member a i of each class to the language and assume that every conjugacy class is ∅-definable.

(2) We may assume every proper normal subgroup of G to be central.

We claim that a normal subgroup must be central or have finite index in G : a normal subgroup is the union of conjugacy classes, hence is ∅-definable. By Lemma 1.1, the conjugacy class of a non central element, a G 1 say, must have maximal Cantor rank over ∅. It follows from Lemma 1.1(3) that any proper infinite normal subgroup has index at most n. One may replace G by a minimal union C of conjugacy classes (with at least one of them non-central) closed under multiplication : as the group C has finite index in G, every possible non-central proper normal subgroup H in C has finite index in G, and would give birth to a subgroup N of H, normal in G, and of finite index in G, contradicting the minimality of C.

(3) We may assume that the centre of G is trivial.

Should G/Z(G) be abelian, G/Z(G) would be be finite, as G has only finitely many conjugacy classes. This is not possible as G is infinite. It follows that the second centre Z 2 (G) of G is a proper normal subgroup in G. By (2), one has Z 2 (G) = Z(G). Moding out by the centre (which preserves weak smallness as well as the assumption that the centralisers have bounded size), we may assume that the centre of G is trivial. (4) The group G is not locally finite.

Assume that G be locally finite. Since it has finite exponent, there is a prime number p such that for every natural number n, there is a finite subgroup H of G whose cardinality is divisible by p n . Then H has Sylow subgroup S of cardinality at least p n . But S has a non-trivial centre, the centraliser of any element of which contains the whole Sylow, a contradiction. Thus, one can consider a finitely generated infinite algebraic closure Γ.

(5) The group Γ has finitely many conjugacy classes.

Any x in Γ can be written a y i . As C(a i ) is finite, y is algebraic over a i and x. (6) One may assume the proper normal subgroups of Γ to be trivial. By ( 2) and ( 3), no proper union of conjugacy classes C 1 , . . . , C m (in the sense of G) is closed under multiplication. We may add finitely many parameters witnessing this fact to the language. We use an argument of Poizat in [START_REF] Poizat | Quelques tentatives de définir une notion générale de groupes et de corps de dimension un et de déterminer leurs propriétés algébriques[END_REF], which we shall call Poizat's symmetry argument. Let a = a i and b = a j be representatives of any two non-trivial conjugacy classes (in particular, a, b are in Γ). For every conjugate xbx -1 of b except a set of small Cantor rank over a and b, the elements axbx -1 and b are conjugates. As a surjection with bounded fibres preserves the rank, for all x except a set of small rank, axbx -1 and b are conjugates. Symmetrically, for all x except a set of small rank, x -1 axb and a are conjugates : one can find some x such that axbx -1 and x -1 axb are conjugated respectively to b and a. Thus, b and a lie in the same conjugacy class.

(9) Final contradiction. G is an infinite group with bounded exponent and only one non-trivial conjugacy class. Such a group does not exist [START_REF] Reineke | Minimale Gruppen[END_REF][START_REF] Poizat | Quelques tentatives de définir une notion générale de groupes et de corps de dimension un et de déterminer leurs propriétés algébriques[END_REF]Reineke]. For instance, as a group of exponent 2 is abelian, the group should have exponent a prime p = 2. If x = 1, the elements x and x -1 would be conjugated under some element y of order 2 modulo the centraliser of x, which prevents the group from having exponent p. Theorem 3.4. A small infinite ℵ 0 -saturated group has an infinite abelian subgroup.

Proof. By Proposition 3.3 and saturation, such a group is either abelian, or has an infinite proper centraliser. Iterating, one either ends on an infinite abelian centraliser after finitely many steps or builds an infinite chain of pairwise commuting elements. These elements generate an infinite abelian subgroup.

Appealing to Hall-Kulatilaka-Kargapolov, who use Feit-Thomson's Theorem, one can say much more, and manage without the compactness theorem. Recall Fact 3.5 (Hall-Kulatilaka-Kargapolov [START_REF] Hall | A Property of Locally Finite Groups[END_REF]). An infinite locally finite group has an infinite abelian subgroup. Theorem 3.6. A weakly small infinite group has an infinite abelian subgroup.

Proof. We need just show that any weakly small infinite group is either abelian or has an infinite proper centraliser : if this is the case, iterating, one either gets an infinite abelian centraliser or builds an infinite chain of pairwise commuting elements.

So let G be a non abelian counter-example. Every non central element of G has finite centraliser, and G has a finite centre. The group G cannot have an infinite abelian subgroup. According to Hall-Kulatilaka-Kargapolov, G is not locally finite. By Lemma 1.1(3), an infinite finitely generated subgroup γ splits into finitely many conjugacy classes (in the sense of G). By Lemma 1.1, these classes have maximal Cantor rank over γ. By Proposition 2.5, the almost stabiliser of every such class is a normal subgroup of finite index in γ. After Poizat's symmetry argument, the intersection of almost stabilisers of all conjugacy classes meeting γ consists of a (finite) central subgroup Z together with C γ ∩ γ, where C γ is a conjugacy class in G. It is easy to see that C γ is the same for all finitely generated infinite subgroups γ, so we can denote this unique conjugacy class by C. We conclude that C ∪ Z is a subgroup of G. Replacing G by the later, we are back to the case where all proper centralisers have bounded size, a contradiction with Proposition 3.3.

Remark 3.7. The initial proof of Theorem 3.4 used Hall-Kulatilaka-Kargapolov. The author is grateful to Poizat who adapted the proof to a weakly small group and made clarifying remarks. Remark 3.8. One cannot expect the infinite abelian group to be definable, as Plotkin found infinite ℵ 0 -categorical groups without infinite definable abelian subgroups [START_REF] Plotkin | ZF and locally finite groups[END_REF].

Small nilpotent groups

We now switch to small nilpotent groups. Let us first recall that the structure of small abelian pure groups is already known : Fact 4.1. (Wagner [START_REF] Frank | Quasi-endomorphisms in small stable groups[END_REF]) A small abelian group is the direct sum of a definable divisible group with one of bounded exponent. Remark 4.2. The group of bounded exponent need not be definable, but it is contained in a definable group of bounded exponent.

Remark 4.3. Since Prüfer and Baer, one knows that a divisible abelian group is isomorphic to direct sums of copies of Q and Prüfer groups, whereas an abelian group of bounded exponent is isomorphic to a direct sum of cyclic groups [START_REF] Fuchs | Infinite abelian groups[END_REF]. It follows that the theory of a small pure group has countably many denumerable pairwise non-isomorphic models ; thus, Vaught's conjecture holds for the theory of a pure abelian group. More generally, Vaught's conjecture holds for every complete first order theory of module over a countable Dedekind ring (and thus for a module over Z), as well as for several classes of modules over countable rings [START_REF] Puninskaya | Vaught's conjecture[END_REF]Puninskaya].

Remark 4.4. Fact 4.1 does not hold for a weakly small abelian group : consider the sum over n of cyclic groups of order p n . But one may say : Proposition 4.5. In a weakly small abelian group, for every natural number n, any element is the sum of an n-divisible element with one of finite order.

Proof. For a contradiction, let us suppose that there be an element x and a natural number n such that xz / ∈ G n for any z having finite order. If there is some y in G and some natural number k such that x kn = y (k+1)n , this yields x = y n (y -n x) with (y -n x) kn = 1, a contradiction. Then, for every natural number k, one has

x kn ∈ G kn \ G (k+1)n . This implies that the chain G ∩ acl(x) > G n ∩ acl(x) > G 2n ∩ acl(x) > • •
• is strictly decreasing and contradicts the weakly small chain condition.

In an abelian group, every divisible group is a direct summand [2, Theorem 1]. This may not be true for a central divisible subgroup of an arbitrary group, even if the ambient group is nilpotent. For instance, consider the subgroup of GL 3 (C) the elements of which are upper triangular matrices with 1 entries on the main diagonal ; it is a nilpotent group whose centre Z is divisible, isomorphic to C × , but Z is no direct summand. However, we claim the following : Proposition 4.6. Let G be a group, and D a divisible subgroup of the centre. There exists a subset A of G, invariant under conjugation and containing every power of its elements, with in addition

G = D • A and D ∩ A = {1} Proof. If A 1 ⊂ A 2 ⊂ • • •
is an increasing chain of subsets each of which contains all its powers and such that A i ∩ D is trivial, then A i still contains all its powers and A i ∩ D is trivial too. By Zorn's Lemma there is a maximal subset A with these properties. We show that D • A equals G. Otherwise, there exists an x not in D • A. By maximality of A, there is a natural number n greater than 1, and some d in D so that x n equals d. We may choose n minimal with this property. Let e be an nth root of d -1 in D, and let y equal xe. Then y n equals one, and y is not in D • A. But the set of powers of y intersects D by maximality of A : there is some natural number m < n such that y m lie in D, and so does x m , a contradiction with the choice of n.

In [START_REF] Nesin | Poly-separated and ω-stable nilpotent groups[END_REF]Nesin], it is shown that an omega-stable nilpotent group is the central product of a definable group with one of bounded exponent. We show that this also holds for a small nilpotent group. Recall that a group G is the central product of two of its normal subgroups, if it is the product of these subgroups and if moreover their intersection lies in the centre of G. For a group G and a subset A of G, we shall write A n for the set of the nth-powers of A, and G ′ for the derived subgroup of G. The following algebraic facts about nilpotent groups can be found in [START_REF] Borovik | Groups of finite Morley rank[END_REF]Chapter 1]. On the other hand, the centre is the sum of a divisible subgroup D 0 with a subgroup F 0 of finite exponent, say l. So C lm is included in D 0 . By Proposition 4.6, there is some set A closed under power operation, such that C = D 0 • A and D 0 ∩ A = {1} ; but A lm is included in D 0 ∩ A, so A has finite exponent, and

G = D • Z(G) • D 0 • A = (D • D 0 ) • (F 0 • A)
Note that since we have D 0 ⊂ G n = D, we get G = D • B where B is a set having finite exponent. Let F be the group generated by B. The abelian group F/F ′ is generated by (B • F ′ )/F ′ and has bounded exponent. Fact 4.8 implies that F has bounded exponent. Theorem 4.10. A small nilpotent group is the central product of a definable divisible group with a definable one of bounded exponent.

Proof. If G is a small abelian group, it is the direct product of a divisible definable group D and of one group F of finite exponent n by Fact 4.1. So it is the product of D and the definable group of every elements of order n. By induction on the nilpotency class, if G is nilpotent of class c + 1, then G/Z(G) is the central sum of some divisible definable normal subgroup A/Z(G) and some group B/Z(G) of finite exponent n. Besides, Z(G) equals D 0 ⊕ F 0 where F 0 has exponent m and D 0 is definable and divisible. We write D for A 2m • D 0 .

Claim. D is a definable divisible normal subgroup of G.

Proof of Claim. Let x be an element in A and q a natural number. As As A/Z(G) is divisible there is some y in A with x -1 y q in Z(G). Then x -2m (y 2m ) q is in D 0 . As D 0 is central and divisible, this proves that A 2m • D 0 is a divisible part. Let us show that it is a group. Let a, b be in A. As A/Z(G) is normal in G/Z(G), there exists a central element z such that ab = baz. Moreover, we have z = d 0 f 0 for some d 0 in D 0 and f m 0 = 1. We obtain

a 2m b 2m = (ab) 2m z (2m-1)+(2m-2)+•••+1 = (ab) 2m z m(2m-1) = (ab) 2m d m(2m-1) 0
A is a subgroup of G so ab is in A, and a 2m b 2m belongs to A 2m • D 0 . A similar argument shows that D is normal in G.

By Fact 4.7, the set G 2mn is included in D, so we may apply Proposition 4.9 : there is a group B of bounded exponent p such that G = D • B. We may assume B to be definable and normal by replacing it with the set {x ∈ G : x p = 1} (the fact that {x ∈ G : x p = 1} is a normal subgroup of G follows from Fact 4.7).

Groups in a small and simple theory

We shall not define what a simple structure is, but refer the interested reader to [START_REF] Frank | Simple Theories[END_REF]Wagner]. We just recall the uniform descending chain condition up to finite index in a group with simple theory : Fact 5.1. (Wagner [25, Theorem 4.2.12]) In a group with simple theory, a descending chain of intersections of a family H 1 , H 2 . . . of subgroups defined respectively by formulae f (x, a 1 ), f (x, a 2 ) . . . where f (x, y) is a fixed formula, becomes stationary after finitely many steps, up to finite index.

Recall that two subgroups of a given group are commensurable if the index of their intersection is finite in both of them. Fact 5.2. (Schlichting [START_REF] Schlichting | Operationen mit periodischen Stabilisatoren[END_REF][START_REF] Frank | Simple Theories[END_REF]) Let G be a group and H a family of uniformly commensurable subgroups. There is a subgroup N of G commensurable with members of H and invariant under the action of the automorphisms group of G stabilising the family H setwise. The inclusions H∈H ⊂ N ⊂ H 4 hold. Moreover, N is a finite extension of a finite intersection of elements in H. In particular, if H consists of definable groups then N is also definable.

We now turn to small simple groups. The first step towards the existence of a definable finite-by-abelian subgroup is to appeal to Theorem 3.4. Note that in a stable group, every set of pairwise commuting elements is trivially contained in a definable abelian subgroup. Shelah showed that this also holds in a dependent group [START_REF] Shelah | Dependent first order theories, continued[END_REF]. The second step is the following : Proposition 5.3. In a group with simple theory, every abelian subgroup A is contained in an A-definable finite-by-abelian subgroup.

Proof. Let G be this group and C a sufficiently saturated elementary extension of G. We work inside C. By Fact 5.

Z * (H) = {h ∈ H : D C (C H (h), ϕ, k) ≥ D C (H, ϕ, k), ϕ formula, k natural number}
Recall that for a partial type π(x, A), the sentence "D C (π(x, A), ϕ, k) ≥ n" is a type-definable condition on A as stated in [25, Remark 4.1.5], so the group Z * (H) is type-definable. By compactness and saturation, centralisers of elements in Z * (H) have bounded index in H, and conjugacy classes in Z * (H) are finite of bounded size. The first observation implies that Z * (H) is definable, and the second one together with [START_REF] Neumann | Groups covered by permutable subsets[END_REF]Theorem 3.1] show that the derived subgroup of Z * (H) is finite. Note that H and Z * (H) are A-definable, hence Z * (H) computed in G fulfills our purpose.

Corollary 5.4. A weakly small infinite group the theory of which is simple has an infinite definable finite-by-abelian subgroup.

Proof. Follows from Theorem 3.6 and Proposition 5.3.

Remark 5.5. Corollary 5.4 states the best possible result as there are ℵ 0 -categorical simple groups without infinite abelian definable subgroups. For instance, infinite extra-special groups of exponent p are ℵ 0 -categorical [START_REF] Felgner | On ℵ 0 -categorical extra-special p-groups[END_REF]Felgner], and supersimple of SU -rank 1 as they can be interpreted in an infinite dimensional vector space over F p endowed with a non degenerate skew-symmetric bilinear form. They have no infinite definable abelian subgroup by [START_REF] Plotkin | ZF and locally finite groups[END_REF]Plotkin].

Corollary 5.6. A weakly small supersimple group of SU -rank 1 is finite-by-abelianby-finite.

As noticed by Aldama in his thesis [START_REF] De | Chaînes et Dépendance[END_REF], Shelah's result concerning abelian subsets of a dependent group extends to a nilpotent subset of a dependent group. Actually Aldama also shows that in a dependent group G any solvable group A is surrounded by a definable solvable group of same derived length, provided that A be normal in G. We are interested in analogues of these results in the context of a group with simple theory. We propose the following definition : Definition 5.7. A group G is almost solvable of class n for some natural number n, if there is a decreasing sequence of subgroups G i such that

G 0 = G G ′ G 1 G ′ 1 • • • G ′ n G n+1 = {1}
and such that the index [G ′ i : G i+1 ] is finite, with G i being normal in G for all i. We call the sequence G i an almost derived series.

An almost solvable group of class zero is a finite-by-abelian group. We may write H ≤ f G to mean that H ≤ G and [G : H] is finite.

Corollary 5.8. In a group with simple theory, let A be a solvable subgroup of derived length n. If A is a normal subgroup, There is an A-definable almost solvable group of class n containing A such that the members of the almost derived series are A-definable.

Proof. Let us show it by induction on the derived length n. Without loss of generality, we may work in a monster model C extending the ambient group. When n equals 0, this is Proposition 5.3. Suppose that the result holds until n -1. By induction hypothesis, there is an A-definable almost solvable group G of derived length n -1 containing A ′ with an A-definable almost derived series, meaning that there are A-definable normal subgroups G 0 , . . . , G n of G such that

G = G 0 G ′ f G 1 • • • G ′ n-1 f G n = {1}
We shall now use an argument of Wagner in [START_REF] Kaplan | extensions in dependent and simple fields[END_REF].

Put P = G 0 × G 1 • • • × G n ≤ C × • • • × C. By Fact
H) contains A, the elements of H are uniformly commensurable. We may now apply Fact 5.2, and be able to find an A-definable group P A ≤ C × • • • × C commensurable with H and by conjugation under elements of A. We enumerate the coordinates of P from 0 to n, we write π k the projection on the kth coordinate and put P k = π k (P A ) for every k in {0, . . . , n}. As P A is a finite extension of a finite intersection I of elements in H, the group P 0 contains A ′ . We still have

P 0 P ′ 0 f P 1 • • • P ′ n-1 f P n = {1}
We claim that every P k is again normal in P 0 . As G 0 normalises P , the group π 0 (H) normalises H. As A is a normal subgroup of C, it follows that conjugations by elements from π 0 (H) stabilise H setwise. By Fact 5.2, π 0 (H) normalises P A . Let a be in A. Similarly, π 0 (H a ) normalises P A . As P 0 ⊂ π 0 (H) 4 by Fact 5.2, P 0 normalises P A hence P k for every k. Now P 0 contains A ′ so the the group P 0 A/P 0 is abelian. According to Proposition 5.3, there is an A-definable group M such that

P 0 A/P 0 ≤ M/P 0 ≤ n i=0
N G (P i ) /P 0 where M ′ /P 0 is finite. Thus, M is the desired almost solvable group of derived length n.

Question. We may define a group G to be almost nilpotent of class n if there exists a decreasing sequence of subgroups G i such that

G 0 = G [G, G 0 ] G 1 [G, G 1 ] G 2 • • • [G, G n ] G n+1 = {1}
and such that the index [[G, G i ] : G i+1 ] is finite and G i is normal in G for all i.

In a group with simple theory, is any nilpotent subgroup of class n contained in a definable almost nilpotent group of class n?

( 7 )

 7 For every conjugacy class a G , the group Stab Γ (a G ) equals Γ. The local stabiliser of a G in Γ is a normal subgroup of Γ by Corollary 2.4. It must be non-trivial according to Proposition 2.5, hence equals Γ by[START_REF] Cherlin | Groups of small Morley rank[END_REF].

( 8 )

 8 G has only one non-central conjugacy class.

Fact 4 . 7 .

 47 In a nilpotent group, any divisible subgroup commutes with elements of finite order.

Fact 4 . 8 .

 48 Let G be a nilpotent group of nilpotent class c. If G/G ′ has exponent n, the exponent of G is a natural number dividing n c . Proposition 4.9. Let G be a nilpotent small group, and D a divisible subgroup containing G n for some non-zero natural number n. Then G equals the product D • F where the group F has bounded exponent. Proof. Note that since D is divisible and G n ⊂ D, we get G n = D. By induction on the nilpotency class of G. If G is abelian, Baer's Theorem [2] concludes. Suppose that the result holds for any small nilpotent group of class c, and that G is nilpotent of class c + 1, and let Z(G) be the centre of G. The group G/Z(G) is nilpotent of class c. The quotient (D • Z(G))/Z(G) is a divisible subgroup and contains G/Z(G) n . By induction hypothesis, G/Z(G) equals the product D•Z(G)/Z(G) • C/Z(G) with C/Z(G) of finite exponent, say m.

  then either CB(H 2 ) < CB(H 1 ), or dCB(H 2 ) < dCB(H 1 ). (2) If H 1 and H 2 have the same Cantor rank, then H 2 ∩ acl(∅) has finite index in H 1 ∩ acl(∅). Proof. If b is an element of acl(∅) in H 1 \ H 2 , the set bH 2 is definable without parameters, and is disjoint from H 2 . This proves the first point. If H 1 and H 2 have the same Cantor rank, one has

  3.A property of weakly small groups Proposition 3.1. An infinite group whose centre has infinite index, and with only one non-central conjugacy class, is not weakly small. Remark 3.2. This is the analogue of the stable case[16, Théorème 3.10] stating that an infinite group with only one non-trivial conjugacy class is unstable, which itself comes from the minimal case[START_REF] Reineke | Minimale Gruppen[END_REF] Reineke].

  1, there exists a finite intersection H of centralisers of elements in A such that H is minimal up to finite index. The group H contains A, and the centraliser of every element in A has finite index in H. Consider the almost centre Z * (H) of H consisting of elements in H the centraliser of which has finite index in H. We claim that Z * (H) is a definable group. It is a subgroup containing A. According to [25, Lemma 4.1.15], a definable subgroup B of C has finite index in C if and only if the equality D C (B, ϕ, k) = D C (C, ϕ, k) holds for every formula ϕ and natural number k. So we have the following equality

  5.1 there is a finite intersection H of A-conjugates of P which is minimal up to finite index. Let us write H for the set of A-conjugates of H. We claim that the elements of H uniformly commensurable. To see that, we consider the almost normaliser {g ∈ C : H g and H are commensurable} of H in C. We write it N * C (H). By [25, Lemma 4.1.15], we have : N * C (H) = {g ∈ C : D C (H ∩ H g , ϕ, k) ≥ D C (H, ϕ, k), ϕ formula, k natural number} It follows from [25, Remark 4.1.5] that N * C (H) is an A-type-definable group. By compactness and saturation, two N * C (H)-conjugates of H are uniformly commensurable. N * C (H) is in fact a definable group. As N * C