text:
Abstraction

Abstraction is a process wherein general rules and concepts are derived from the usage and classification of specific examples, literal (real or concrete) signifiers, first principles, or other methods.

"An abstraction" is the outcome of this process—a concept that acts as a common noun for all subordinate concepts and connects any related concepts as a group, field, or category.

Conceptual abstractions may be formed by filtering the information content of a concept or an observable phenomenon, selecting only those aspects which are relevant for a particular purpose. For example, abstracting a leather soccer ball to the more general idea of a ball selects only the information on general ball attributes and behavior, excluding but not eliminating the other phenomenal and cognitive characteristics of that particular ball. In a type–token distinction, a type (e.g., a 'ball') is more abstract than its tokens (e.g., 'that leather soccer ball').

Abstraction in its secondary use is a material process, discussed in the themes below.

Origins
Thinking in abstractions is considered by anthropologists, archaeologists, and sociologists to be one of the key traits in modern human behaviour, which is believed to have developed between 50,000 and 100,000 years ago. Its development is likely to have been closely connected with the development of human language, which (whether spoken or written) appears to both involve and facilitate abstract thinking.

History
Abstraction involves induction of ideas or the synthesis of particular facts into one general theory about something. It is the opposite of specification, which is the analysis or breaking-down of a general idea or abstraction into concrete facts. Abstraction can be illustrated by Francis Bacon's Novum Organum (1620), a book of modern scientific philosophy written in the late Jacobean era of England to encourage modern thinkers to collect specific facts before making any generalizations.

Bacon used and promoted induction as an abstraction tool; it complemented but was distinct from the ancient deductive-thinking approach that had dominated the intellectual world since the times of Greek philosophers like Thales, Anaximander, and Aristotle. Thales (c. 624–546 BCE) believed that everything in the universe comes from one main substance, water. He deduced or specified from a general idea, "everything is water," to the specific forms of water such as ice, snow, fog, and rivers.

Modern scientists used the approach of abstraction (going from particular facts collected into one general idea). Newton (1642–1727) derived the motion of the planets from Copernicus' (1473–1543) simplification, that the Sun is the center of the Solar System; Kepler (1571–1630) compressed thousands of measurements into one expression to finally conclude that Mars moves in an elliptical orbit about the Sun; Galileo (1564–1642) repeated one hundred specific experiments into the law of falling bodies.

Compression
An abstraction can be seen as a compression process, mapping multiple different pieces of constituent data to a single piece of abstract data; based on similarities in the constituent data, for example, many different physical cats map to the abstraction "CAT". This conceptual scheme emphasizes the inherent equality of both constituent and abstract data, thus avoiding problems arising from the distinction between "abstract" and "concrete". In this sense the process of abstraction entails the identification of similarities between objects, and the process of associating these objects with an abstraction (which is itself an object).
 * For example, picture 1 below illustrates the concrete relationship "Cat sits on Mat".

Chains of abstractions can be construed, moving from neural impulses arising from sensory perception to basic abstractions such as color or shape, to experiential abstractions such as a specific cat, to semantic abstractions such as the "idea" of a CAT, to classes of objects such as "mammals" and even categories such as "object" as opposed to "action".
 * For example, graph 1 below expresses the abstraction "agent sits on location". This conceptual scheme entails no specific hierarchical taxonomy (such as the one mentioned involving cats and mammals), only a progressive exclusion of detail.

Instantiation
Non-existent things in any particular place and time are often seen as abstract. By contrast, instances, or members, of such an abstract thing might exist in many different places and times.

Those abstract things are then said to be multiply instantiated, in the sense of picture 1, picture 2, etc., shown below. It is not sufficient, however, to define abstract ideas as those that can be instantiated and to define abstraction as the movement in the opposite direction to instantiation. Doing so would make the concepts "cat" and "telephone" abstract ideas since despite their varying appearances, a particular cat or a particular telephone is an instance of the concept "cat" or the concept "telephone". Although the concepts "cat" and "telephone" are abstractions, they are not abstract in the sense of the objects in graph 1 below. We might look at other graphs, in a progression from cat to mammal to animal, and see that animal is more abstract than mammal; but on the other hand mammal is a harder idea to express, certainly in relation to marsupial or monotreme.

Perhaps confusingly, some philosophies refer to tropes (instances of properties) as abstract particulars—e.g., the particular redness of a particular apple is an abstract particular. This is similar to qualia and sumbebekos.

Material process
Still retaining the primary meaning of 'abstrere' or 'to draw away from', the abstraction of money, for example, works by drawing away from the particular value of things allowing completely incommensurate objects to be compared (see the section on 'Physicality' below). Karl Marx's writing on the commodity abstraction recognizes a parallel process.

The state (polity) as both concept and material practice exemplifies the two sides of this process of abstraction. Conceptually, 'the current concept of the state is an abstraction from the much more concrete early-modern use as the standing or status of the prince, his visible estates'. At the same time, materially, the 'practice of statehood is now constitutively and materially more abstract than at the time when princes ruled as the embodiment of extended power'.

Ontological status
The way that physical objects, like rocks and trees, have being differs from the way that properties of abstract concepts or relations have being, for example the way the concrete, particular, individuals pictured in picture 1 exist differs from the way the concepts illustrated in graph 1 exist. That difference accounts for the ontological usefulness of the word "abstract". The word applies to properties and relations to mark the fact that, if they exist, they do not exist in space or time, but that instances of them can exist, potentially in many different places and times.

Physicality
A physical object (a possible referent of a concept or word) is considered concrete (not abstract) if it is a particular individual that occupies a particular place and time. However, in the secondary sense of the term 'abstraction', this physical object can carry materially abstracting processes. For example, record-keeping aids throughout the Fertile Crescent included calculi (clay spheres, cones, etc.) which represented counts of items, probably livestock or grains, sealed in containers. According to, these clay containers contained tokens, the total of which were the count of objects being transferred. The containers thus served as something of a bill of lading or an accounts book. In order to avoid breaking open the containers for the count, marks were placed on the outside of the containers. These physical marks, in other words, acted as material abstractions of a materially abstract process of accounting, using conceptual abstractions (numbers) to communicate its meaning.

Abstract things are sometimes defined as those things that do not exist in reality or exist only as sensory experiences, like the color red. That definition, however, suffers from the difficulty of deciding which things are real (i.e. which things exist in reality). For example, it is difficult to agree to whether concepts like God, the number three, and goodness are real, abstract, or both.

An approach to resolving such difficulty is to use predicates as a general term for whether things are variously real, abstract, concrete, or of a particular property (e.g., good). Questions about the properties of things are then propositions about predicates, which propositions remain to be evaluated by the investigator. In the graph 1 below, the graphical relationships like the arrows joining boxes and ellipses might denote predicates.

Referencing and referring
Abstractions sometimes have ambiguous referents. For example, "happiness" can mean experiencing various positive emotions, but can also refer to life satisfaction and subjective well-being. Likewise, "architecture" refers not only to the design of safe, functional buildings, but also to elements of creation and innovation which aim at elegant solutions to construction problems, to the use of space, and to the attempt to evoke an emotional response in the builders, owners, viewers and users of the building.

Simplification and ordering
Abstraction uses a strategy of simplification, wherein formerly concrete details are left ambiguous, vague, or undefined; thus effective communication about things in the abstract requires an intuitive or common experience between the communicator and the communication recipient. This is true for all verbal/abstract communication. For example, many different things can be red. Likewise, many things sit on surfaces (as in picture 1, to the right). The property of redness and the relation sitting-on are therefore abstractions of those objects. Specifically, the conceptual diagram graph 1 identifies only three boxes, two ellipses, and four arrows (and their five labels), whereas the picture 1 shows much more pictorial detail, with the scores of implied relationships as implicit in the picture rather than with the nine explicit details in the graph.

Graph 1 details some explicit relationships between the objects of the diagram. For example, the arrow between the agent and CAT:Elsie depicts an example of an is-a relationship, as does the arrow between the location and the MAT. The arrows between the gerund/present participle SITTING and the nouns agent and location express the diagram's basic relationship; "agent is SITTING on location"; Elsie is an instance of CAT.

Although the description sitting-on (graph 1) is more abstract than the graphic image of a cat sitting on a mat (picture 1), the delineation of abstract things from concrete things is somewhat ambiguous; this ambiguity or vagueness is characteristic of abstraction. Thus something as simple as a newspaper might be specified to six levels, as in Douglas Hofstadter's illustration of that ambiguity, with a progression from abstract to concrete in Gödel, Escher, Bach (1979): ":(1) a publication
 * (2) a newspaper
 * (3) The San Francisco Chronicle
 * (4) the May 18 edition of The San Francisco Chronicle
 * (5) my copy of the May 18 edition of The San Francisco Chronicle
 * (6) my copy of the May 18 edition of The San Francisco Chronicle as it was when I first picked it up (as contrasted with my copy as it was a few days later: in my fireplace, burning)"

An abstraction can thus encapsulate each of these levels of detail with no loss of generality. But perhaps a detective or philosopher/scientist/engineer might seek to learn about something, at progressively deeper levels of detail, to solve a crime or a puzzle.

Thought processes
In philosophical terminology, abstraction is the thought process wherein ideas are distanced from objects. But an idea can be symbolized.

In art
Typically, abstraction is used in the arts as a synonym for abstract art in general. Strictly speaking, it refers to art unconcerned with the literal depiction of things from the visible world—it can, however, refer to an object or image which has been distilled from the real world, or indeed, another work of art. Artwork that reshapes the natural world for expressive purposes is called abstract; that which derives from, but does not imitate a recognizable subject is called nonobjective abstraction. In the 20th century the trend toward abstraction coincided with advances in science, technology, and changes in urban life, eventually reflecting an interest in psychoanalytic theory. Later still, abstraction was manifest in more purely formal terms, such as color, freedom from objective context, and a reduction of form to basic geometric designs.

In computer science
Computer scientists use abstraction to make models that can be used and re-used without having to re-write all the program code for each new application on every different type of computer. They communicate their solutions with the computer by writing source code in some particular computer language which can be translated into machine code for different types of computers to execute. Abstraction allows program designers to separate a framework (categorical concepts related to computing problems) from specific instances which implement details. This means that the program code can be written so that code does not have to depend on the specific details of supporting applications, operating system software, or hardware, but on a categorical concept of the solution. A solution to the problem can then be integrated into the system framework with minimal additional work. This allows programmers to take advantage of another programmer's work, while requiring only an abstract understanding of the implementation of another's work, apart from the problem that it solves.

In general semantics
Abstractions and levels of abstraction play an important role in the theory of general semantics originated by Alfred Korzybski. Anatol Rapoport wrote "Abstracting is a mechanism by which an infinite variety of experiences can be mapped on short noises (words)."

In history
Francis Fukuyama defines history as "a deliberate attempt of abstraction in which we separate out important from unimportant events".

In linguistics
Researchers in linguistics frequently apply abstraction so as to allow an analysis of the phenomena of language at the desired level of detail. A commonly used abstraction, the phoneme, abstracts speech sounds in such a way as to neglect details that cannot serve to differentiate meaning. Other analogous kinds of abstractions (sometimes called "emic units") considered by linguists include morphemes, graphemes, and lexemes.

Abstraction also arises in the relation between syntax, semantics, and pragmatics. Pragmatics involves considerations that make reference to the user of the language; semantics considers expressions and what they denote (the designata) abstracted from the language user; and syntax considers only the expressions themselves, abstracted from the designate.

In mathematics
Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept or object, removing any dependence on real-world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.

The advantages of abstraction in mathematics are:
 * It reveals deep connections between different areas of mathematics.
 * Known results in one area can suggest conjectures in another related area.
 * Techniques and methods from one area can be applied to prove results in other related area.
 * Patterns from one mathematical object can be generalized to other similar objects in the same class.

The main disadvantage of abstraction is that highly abstract concepts are more difficult to learn, and might require a degree of mathematical maturity and experience before they can be assimilated.

In music
In music, the term abstraction can be used to describe improvisatory approaches to interpretation, and may sometimes indicate abandonment of tonality. Atonal music has no key signature, and is characterized by the exploration of internal numeric relationships.

In neurology
A recent meta-analysis suggests that the verbal system has a greater engagement with abstract concepts when the perceptual system is more engaged in processing concrete concepts. This is because abstract concepts elicit greater brain activity in the inferior frontal gyrus and middle temporal gyrus compared to concrete concepts which elicit greater activity in the posterior cingulate, precuneus, fusiform gyrus, and parahippocampal gyrus. Other research into the human brain suggests that the left and right hemispheres differ in their handling of abstraction. For example, one meta-analysis reviewing human brain lesions has shown a left hemisphere bias during tool usage.

In philosophy
Abstraction in philosophy is the process (or, to some, the alleged process) in concept formation of recognizing some set of common features in individuals, and on that basis forming a concept of that feature. The notion of abstraction is important to understanding some philosophical controversies surrounding empiricism and the problem of universals. It has also recently become popular in formal logic under predicate abstraction. Another philosophical tool for the discussion of abstraction is thought space.

John Locke defined abstraction in An Essay Concerning Human Understanding:

'So words are used to stand as outward marks of our internal ideas, which are taken from particular things; but if every particular idea that we take in had its own special name, there would be no end to names. To prevent this, the mind makes particular ideas received from particular things become general; which it does by considering them as they are in the mind—mental appearances—separate from all other existences, and from the circumstances of real existence, such as time, place, and so on. This procedure is called abstraction. In it, an idea taken from a particular thing becomes a general representative of all of the same kind, and its name becomes a general name that is applicable to any existing thing that fits that abstract idea.' (2.11.9)

In psychology
Carl Jung's definition of abstraction broadened its scope beyond the thinking process to include exactly four mutually exclusive, different complementary psychological functions: sensation, intuition, feeling, and thinking. Together they form a structural totality of the differentiating abstraction process. Abstraction operates in one of these functions when it excludes the simultaneous influence of the other functions and other irrelevancies, such as emotion. Abstraction requires selective use of this structural split of abilities in the psyche. The opposite of abstraction is concretism. Abstraction is one of Jung's 57 definitions in Chapter XI of Psychological Types. "There is an abstract thinking, just as there is abstract feeling, sensation and intuition. Abstract thinking singles out the rational, logical qualities ... Abstract feeling does the same with ... its feeling-values. ... I put abstract feelings on the same level as abstract thoughts. ... Abstract sensation would be aesthetic as opposed to sensuous sensation and abstract intuition would be symbolic as opposed to fantastic intuition. (Jung, [1921] (1971): par. 678)."

In social theory
Social theorists deal with abstraction both as an ideational and as a material process. Alfred Sohn-Rethel (1899–1990) asked: "Can there be abstraction other than by thought?" He used the example of commodity abstraction to show that abstraction occurs in practice as people create systems of abstract exchange that extend beyond the immediate physicality of the object and yet have real and immediate consequences. This work was extended through the 'Constitutive Abstraction' approach of writers associated with the Journal Arena. Two books that have taken this theme of the abstraction of social relations as an organizing process in human history are Nation Formation: Towards a Theory of Abstract Community (1996) and an associated volume published in 2006, Globalism, Nationalism, Tribalism: Bringing Theory Back In. These books argue that a nation is an abstract community bringing together strangers who will never meet as such; thus constituting materially real and substantial, but abstracted and mediated relations. The books suggest that contemporary processes of globalization and mediatization have contributed to materially abstracting relations between people, with major consequences for how humans live their lives.

One can readily argue that abstraction is an elementary methodological tool in several disciplines of social science. These disciplines have definite and different concepts of "man" that highlight those aspects of man and his behaviour by idealization that are relevant for the given human science. For example, homo sociologicus is the man as sociology abstracts and idealizes it, depicting man as a social being. Moreover, we could talk about homo cyber sapiens (the man who can extend his biologically determined intelligence thanks to new technologies), or homo creativus (who is simply creative).

Abstraction (combined with Weberian idealization) plays a crucial role in economics - hence abstractions such as "the market" and the generalized concept of "business". Breaking away from directly experienced reality was a common trend in 19th-century sciences (especially physics), and this was the effort which fundamentally determined the way economics tried (and still tries) to approach the economic aspects of social life. It is abstraction we meet in the case of both Newton's physics and the neoclassical theory, since the goal was to grasp the unchangeable and timeless essence of phenomena. For example, Newton created the concept of the material point by following the abstraction method so that he abstracted from the dimension and shape of any perceptible object, preserving only inertial and translational motion. Material point is the ultimate and common feature of all bodies. Neoclassical economists created the indefinitely abstract notion of homo economicus by following the same procedure. Economists abstract from all individual and personal qualities in order to get to those characteristics that embody the essence of economic activity. Eventually, it is the substance of the economic man that they try to grasp. Any characteristic beyond it only disturbs the functioning of this essential core.

title: Abstraction


text:
Abelian group

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.

The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified.

Definition
An abelian group is a set $$A$$, together with an operation $$\cdot$$ that combines any two elements $$a$$ and $$b$$ of $$A$$ to form another element of $$A,$$ denoted $$a \cdot b$$. The symbol $$\cdot$$ is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, $$(A, \cdot)$$, must satisfy four requirements known as the abelian group axioms (some authors include in the axioms some properties that belong to the definition of an operation: namely that the operation is defined for any ordered pair of elements of $A$, that the result is well-defined, and that the result belongs to $A$):

 * Associativity: For all $$a$$, $$b$$, and $$c$$ in $$A$$, the equation $$(a \cdot b)\cdot c = a \cdot (b \cdot c)$$ holds.
 * Identity element: There exists an element $$e$$ in $$A$$, such that for all elements $$a$$ in $$A$$, the equation $$e \cdot a = a \cdot e = a$$ holds.
 * Inverse element: For each $$a$$ in $$A$$ there exists an element $$b$$ in $$A$$ such that $$a \cdot b = b \cdot a = e$$, where $$e$$ is the identity element.
 * Commutativity: For all $$a$$, $$b$$ in $$A$$, $$a \cdot b = b \cdot a$$.

A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group".

Notation
There are two main notational conventions for abelian groups – additive and multiplicative.

Generally, the multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules and rings. The additive notation may also be used to emphasize that a particular group is abelian, whenever both abelian and non-abelian groups are considered, some notable exceptions being near-rings and partially ordered groups, where an operation is written additively even when non-abelian.

Multiplication table
To verify that a finite group is abelian, a table (matrix) – known as a Cayley table – can be constructed in a similar fashion to a multiplication table. If the group is $$G = \{g_1 = e, g_2, \dots, g_n \}$$ under the operation $\cdot$, the $(i, j)$-th entry of this table contains the product $$g_i \cdot g_j$$.

The group is abelian if and only if this table is symmetric about the main diagonal. This is true since the group is abelian iff $$g_i \cdot g_j = g_j \cdot g_i$$ for all $$i, j = 1, ..., n$$, which is iff the $$(i, j)$$ entry of the table equals the $$(j, i)$$ entry for all $$i, j = 1, ..., n$$, i.e. the table is symmetric about the main diagonal.

Examples

 * For the integers and the operation addition $$+$$, denoted $$(\mathbb{Z}, +)$$, the operation + combines any two integers to form a third integer, addition is associative, zero is the additive identity, every integer $$n$$ has an additive inverse, $$-n$$, and the addition operation is commutative since $$n + m = m + n$$ for any two integers $$m$$ and $$n$$.
 * Every cyclic group $$G$$ is abelian, because if $$x$$, $$y$$ are in $$G$$, then $$xy = a^ma^n = a^{m+n} = a^na^m = yx$$. Thus the integers, $$\mathbb{Z}$$, form an abelian group under addition, as do the integers modulo $n$, $$\mathbb{Z}/n \mathbb{Z}$$.
 * Every ring is an abelian group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group. In particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication.
 * Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups are exactly the cyclic groups of prime order.
 * The concepts of abelian group and $$\mathbb{Z}$$-module agree. More specifically, every $$\mathbb{Z}$$-module is an abelian group with its operation of addition, and every abelian group is a module over the ring of integers $$\mathbb{Z}$$ in a unique way.

In general, matrices, even invertible matrices, do not form an abelian group under multiplication because matrix multiplication is generally not commutative. However, some groups of matrices are abelian groups under matrix multiplication – one example is the group of $$2 \times 2$$ rotation matrices.

Historical remarks
Camille Jordan named abelian groups after Norwegian mathematician Niels Henrik Abel, as Abel had found that the commutativity of the group of a polynomial implies that the roots of the polynomial can be calculated by using radicals.

Properties
If $$n$$ is a natural number and $$x$$ is an element of an abelian group $$G$$ written additively, then $$nx$$ can be defined as $$x + x + \cdots + x$$ ($$n$$ summands) and $$(-n)x = -(nx)$$. In this way, $$G$$ becomes a module over the ring $$\mathbb{Z}$$ of integers. In fact, the modules over $$\mathbb{Z}$$ can be identified with the abelian groups.

Theorems about abelian groups (i.e. modules over the principal ideal domain $$\mathbb{Z}$$) can often be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classification of finitely generated abelian groups which is a specialization of the structure theorem for finitely generated modules over a principal ideal domain. In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direct sum of a torsion group and a free abelian group. The former may be written as a direct sum of finitely many groups of the form $$\mathbb{Z}/p^k\mathbb{Z}$$ for $$p$$ prime, and the latter is a direct sum of finitely many copies of $$\mathbb{Z}$$.

If $$f, g: G \to H$$ are two group homomorphisms between abelian groups, then their sum $$f + g$$, defined by $$(f + g)(x) = f(x) + g(x)$$, is again a homomorphism. (This is not true if $$H$$ is a non-abelian group.) The set $$\text{Hom}(G,H)$$ of all group homomorphisms from $$G$$ to $$H$$ is therefore an abelian group in its own right.

Somewhat akin to the dimension of vector spaces, every abelian group has a rank. It is defined as the maximal cardinality of a set of linearly independent (over the integers) elements of the group. Finite abelian groups and torsion groups have rank zero, and every abelian group of rank zero is a torsion group. The integers and the rational numbers have rank one, as well as every nonzero additive subgroup of the rationals. On the other hand, the multiplicative group of the nonzero rationals has an infinite rank, as it is a free abelian group with the set of the prime numbers as a basis (this results from the fundamental theorem of arithmetic).

The center $$Z(G)$$ of a group $$G$$ is the set of elements that commute with every element of $$G$$. A group $$G$$ is abelian if and only if it is equal to its center $$Z(G)$$. The center of a group $$G$$ is always a characteristic abelian subgroup of $$G$$. If the quotient group $$G/Z(G)$$ of a group by its center is cyclic then $$G$$ is abelian.

Finite abelian groups
Cyclic groups of integers modulo $n$, $$\mathbb{Z}/n\mathbb{Z}$$, were among the first examples of groups. It turns out that an arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The automorphism group of a finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper of Georg Frobenius and Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of linear algebra.

Any group of prime order is isomorphic to a cyclic group and therefore abelian. Any group whose order is a square of a prime number is also abelian. In fact, for every prime number $$p$$ there are (up to isomorphism) exactly two groups of order $$p^2$$, namely $$\mathbb{Z}_{p^2}$$ and $$\mathbb{Z}_p\times\mathbb{Z}_p$$.

Classification
The fundamental theorem of finite abelian groups states that every finite abelian group $$G$$ can be expressed as the direct sum of cyclic subgroups of prime-power order; it is also known as the basis theorem for finite abelian groups. Moreover, automorphism groups of cyclic groups are examples of abelian groups. This is generalized by the fundamental theorem of finitely generated abelian groups, with finite groups being the special case when G has zero rank; this in turn admits numerous further generalizations.

The classification was proven by Leopold Kronecker in 1870, though it was not stated in modern group-theoretic terms until later, and was preceded by a similar classification of quadratic forms by Carl Friedrich Gauss in 1801; see history for details.

The cyclic group $$\mathbb{Z}_{mn}$$ of order $$mn$$ is isomorphic to the direct sum of $$\mathbb{Z}_m$$ and $$\mathbb{Z}_n$$ if and only if $$m$$ and $$n$$ are coprime. It follows that any finite abelian group $$G$$ is isomorphic to a direct sum of the form

 * $$\bigoplus_{i=1}^{u}\ \mathbb{Z}_{k_i}$$

in either of the following canonical ways:
 * the numbers $$k_1, k_2, \dots, k_u$$ are powers of (not necessarily distinct) primes,
 * or $$k_1$$ divides $$k_2$$, which divides $$k_3$$, and so on up to $$k_u$$.

For example, $$\mathbb{Z}_{15}$$ can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: $$\mathbb{Z}_{15} \cong \{0,5,10\} \oplus \{0,3,6,9,12\}$$. The same can be said for any abelian group of order 15, leading to the remarkable conclusion that all abelian groups of order 15 are isomorphic.

For another example, every abelian group of order 8 is isomorphic to either $$\mathbb{Z}_8$$ (the integers 0 to 7 under addition modulo 8), $$\mathbb{Z}_4\oplus \mathbb{Z}_2$$ (the odd integers 1 to 15 under multiplication modulo 16), or $$\mathbb{Z}_2\oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$$.

See also list of small groups for finite abelian groups of order 30 or less.

Automorphisms
One can apply the fundamental theorem to count (and sometimes determine) the automorphisms of a given finite abelian group $$G$$. To do this, one uses the fact that if $$G$$ splits as a direct sum $$H\oplus K$$ of subgroups of coprime order, then
 * $$\operatorname{Aut}(H\oplus K) \cong \operatorname{Aut}(H)\oplus \operatorname{Aut}(K).$$

Given this, the fundamental theorem shows that to compute the automorphism group of $$G$$ it suffices to compute the automorphism groups of the Sylow $$p$$-subgroups separately (that is, all direct sums of cyclic subgroups, each with order a power of $$p$$). Fix a prime $$p$$ and suppose the exponents $$e_i$$ of the cyclic factors of the Sylow $$p$$-subgroup are arranged in increasing order:

 * $$e_1\leq e_2 \leq\cdots\leq e_n$$

for some $$n > 0$$. One needs to find the automorphisms of

 * $$\mathbf{Z}_{p^{e_1}} \oplus \cdots \oplus \mathbf{Z}_{p^{e_n}}.$$

One special case is when $$n = 1$$, so that there is only one cyclic prime-power factor in the Sylow $$p$$-subgroup $$P$$. In this case the theory of automorphisms of a finite cyclic group can be used. Another special case is when $$n$$ is arbitrary but $$e_i = 1$$ for $$ 1 \le i \le n$$. Here, one is considering $$P$$ to be of the form

 * $$\mathbf{Z}_p \oplus \cdots \oplus \mathbf{Z}_p,$$

so elements of this subgroup can be viewed as comprising a vector space of dimension $$n$$ over the finite field of $$p$$ elements $$\mathbb{F}_p$$. The automorphisms of this subgroup are therefore given by the invertible linear transformations, so

 * $$\operatorname{Aut}(P)\cong\mathrm{GL}(n,\mathbf{F}_p),$$

where $$\mathrm{GL}$$ is the appropriate general linear group. This is easily shown to have order

 * $$ \left|\operatorname{Aut}(P)\right|=(p^n-1)\cdots(p^n-p^{n-1}).$$

In the most general case, where the $$e_i$$ and $$n$$ are arbitrary, the automorphism group is more difficult to determine. It is known, however, that if one defines

 * $$d_k=\max\{r\mid e_r = e_k\}$$

and

 * $$c_k=\min\{r\mid e_r=e_k\}$$

then one has in particular $$k \le d_k$$, $$c_k \le k$$, and

 * $$ \left|\operatorname{Aut}(P)\right| = \prod_{k=1}^n (p^{d_k}-p^{k-1}) \prod_{j=1}^n (p^{e_j})^{n-d_j} \prod_{i=1}^n (p^{e_i-1})^{n-c_i+1}. $$

One can check that this yields the orders in the previous examples as special cases (see Hillar & Rhea).

Finitely generated abelian groups
An abelian group $A$ is finitely generated if it contains a finite set of elements (called generators) $$G=\{x_1, \ldots, x_n\}$$ such that every element of the group is a linear combination with integer coefficients of elements of $G$.

Let $L$ be a free abelian group with basis $$B=\{b_1, \ldots, b_n\}.$$ There is a unique group homomorphism $$p\colon L \to A,$$ such that
 * $$p(b_i) = x_i\quad \text{for } i=1,\ldots, n.$$

This homomorphism is surjective, and its kernel is finitely generated (since integers form a Noetherian ring). Consider the matrix $M$ with integer entries, such that the entries of its $j$th column are the coefficients of the $j$th generator of the kernel. Then, the abelian group is isomorphic to the cokernel of linear map defined by $M$. Conversely every integer matrix defines a finitely generated abelian group.

It follows that the study of finitely generated abelian groups is totally equivalent with the study of integer matrices. In particular, changing the generating set of $A$ is equivalent with multiplying $M$ on the left by a unimodular matrix (that is, an invertible integer matrix whose inverse is also an integer matrix). Changing the generating set of the kernel of $M$ is equivalent with multiplying $M$ on the right by a unimodular matrix.

The Smith normal form of $M$ is a matrix
 * $$S=UMV,$$

where $U$ and $V$ are unimodular, and $S$ is a matrix such that all non-diagonal entries are zero, the non-zero diagonal entries $d_{1,1}, \ldots, d_{k,k}$ are the first ones, and $d_{j,j}$ is a divisor of $d_{i,i}$ for $i > j$. The existence and the shape of the Smith normal form proves that the finitely generated abelian group $A$ is the direct sum
 * $$\Z^r \oplus \Z/d_{1,1}\Z \oplus \cdots \oplus \Z/d_{k,k}\Z,$$

where $r$ is the number of zero rows at the bottom of $S$ (and also the rank of the group). This is the fundamental theorem of finitely generated abelian groups.

The existence of algorithms for Smith normal form shows that the fundamental theorem of finitely generated abelian groups is not only a theorem of abstract existence, but provides a way for computing expression of finitely generated abelian groups as direct sums.

Infinite abelian groups
The simplest infinite abelian group is the infinite cyclic group $$\mathbb{Z}$$. Any finitely generated abelian group $$A$$ is isomorphic to the direct sum of $$r$$ copies of $$\mathbb{Z}$$ and a finite abelian group, which in turn is decomposable into a direct sum of finitely many cyclic groups of prime power orders. Even though the decomposition is not unique, the number $$r$$, called the rank of $$A$$, and the prime powers giving the orders of finite cyclic summands are uniquely determined.

By contrast, classification of general infinitely generated abelian groups is far from complete. Divisible groups, i.e. abelian groups $$A$$ in which the equation $$nx = a$$ admits a solution $$x \in A$$ for any natural number $$n$$ and element $$a$$ of $$A$$, constitute one important class of infinite abelian groups that can be completely characterized. Every divisible group is isomorphic to a direct sum, with summands isomorphic to $$\mathbb{Q}$$ and Prüfer groups $$\mathbb{Q}_p/Z_p$$ for various prime numbers $$p$$, and the cardinality of the set of summands of each type is uniquely determined. Moreover, if a divisible group $$A$$ is a subgroup of an abelian group $$G$$ then $$A$$ admits a direct complement: a subgroup $$C$$ of $$G$$ such that $$G = A \oplus C$$. Thus divisible groups are injective modules in the category of abelian groups, and conversely, every injective abelian group is divisible (Baer's criterion). An abelian group without non-zero divisible subgroups is called reduced.

Two important special classes of infinite abelian groups with diametrically opposite properties are torsion groups and torsion-free groups, exemplified by the groups $$\mathbb{Q}/\mathbb{Z}$$ (periodic) and $$\mathbb{Q}$$ (torsion-free).

Torsion groups
An abelian group is called periodic or torsion, if every element has finite order. A direct sum of finite cyclic groups is periodic. Although the converse statement is not true in general, some special cases are known. The first and second Prüfer theorems state that if $$A$$ is a periodic group, and it either has a bounded exponent, i.e., $$nA = 0$$ for some natural number $$n$$, or is countable and the $p$-heights of the elements of $$A$$ are finite for each $$p$$, then $$A$$ is isomorphic to a direct sum of finite cyclic groups. The cardinality of the set of direct summands isomorphic to $$\mathbb{Z}/p^m\mathbb{Z}$$ in such a decomposition is an invariant of $$A$$. These theorems were later subsumed in the Kulikov criterion. In a different direction, Helmut Ulm found an extension of the second Prüfer theorem to countable abelian $$p$$-groups with elements of infinite height: those groups are completely classified by means of their Ulm invariants.

Torsion-free and mixed groups
An abelian group is called torsion-free if every non-zero element has infinite order. Several classes of torsion-free abelian groups have been studied extensively:

 * Free abelian groups, i.e. arbitrary direct sums of $$\mathbb{Z}$$
 * Cotorsion and algebraically compact torsion-free groups such as the $p$-adic integers
 * Slender groups

An abelian group that is neither periodic nor torsion-free is called mixed. If $$A$$ is an abelian group and $$T(A)$$ is its torsion subgroup, then the factor group $$A/T(A)$$ is torsion-free. However, in general the torsion subgroup is not a direct summand of $$A$$, so $$A$$ is not isomorphic to $$T(A) \oplus A/T(A)$$. Thus the theory of mixed groups involves more than simply combining the results about periodic and torsion-free groups. The additive group $$\mathbb{Z}$$ of integers is torsion-free $$\mathbb{Z}$$-module.

Invariants and classification
One of the most basic invariants of an infinite abelian group $$A$$ is its rank: the cardinality of the maximal linearly independent subset of $$A$$. Abelian groups of rank 0 are precisely the periodic groups, while torsion-free abelian groups of rank 1 are necessarily subgroups of $$\mathbb{Q}$$ and can be completely described. More generally, a torsion-free abelian group of finite rank $$r$$ is a subgroup of $$\mathbb{Q}_r$$. On the other hand, the group of $p$-adic integers $$\mathbb{Z}_p$$ is a torsion-free abelian group of infinite $$\mathbb{Z}$$-rank and the groups $$\mathbb{Z}_p^n$$ with different $$n$$ are non-isomorphic, so this invariant does not even fully capture properties of some familiar groups.

The classification theorems for finitely generated, divisible, countable periodic, and rank 1 torsion-free abelian groups explained above were all obtained before 1950 and form a foundation of the classification of more general infinite abelian groups. Important technical tools used in classification of infinite abelian groups are pure and basic subgroups. Introduction of various invariants of torsion-free abelian groups has been one avenue of further progress. See the books by Irving Kaplansky, László Fuchs, Phillip Griffith, and David Arnold, as well as the proceedings of the conferences on Abelian Group Theory published in Lecture Notes in Mathematics for more recent findings.

Additive groups of rings
The additive group of a ring is an abelian group, but not all abelian groups are additive groups of rings (with nontrivial multiplication). Some important topics in this area of study are:

 * Tensor product
 * A.L.S. Corner's results on countable torsion-free groups
 * Shelah's work to remove cardinality restrictions
 * Burnside ring

Relation to other mathematical topics
Many large abelian groups possess a natural topology, which turns them into topological groups.

The collection of all abelian groups, together with the homomorphisms between them, forms the category $$\textbf{Ab}$$, the prototype of an abelian category.

proved that the first-order theory of abelian groups, unlike its non-abelian counterpart, is decidable. Most algebraic structures other than Boolean algebras are undecidable.

There are still many areas of current research:
 * Amongst torsion-free abelian groups of finite rank, only the finitely generated case and the rank 1 case are well understood;
 * There are many unsolved problems in the theory of infinite-rank torsion-free abelian groups;
 * While countable torsion abelian groups are well understood through simple presentations and Ulm invariants, the case of countable mixed groups is much less mature.
 * Many mild extensions of the first-order theory of abelian groups are known to be undecidable.
 * Finite abelian groups remain a topic of research in computational group theory.

Moreover, abelian groups of infinite order lead, quite surprisingly, to deep questions about the set theory commonly assumed to underlie all of mathematics. Take the Whitehead problem: are all Whitehead groups of infinite order also free abelian groups? In the 1970s, Saharon Shelah proved that the Whitehead problem is:
 * Undecidable in ZFC (Zermelo–Fraenkel axioms), the conventional axiomatic set theory from which nearly all of present-day mathematics can be derived. The Whitehead problem is also the first question in ordinary mathematics proved undecidable in ZFC;
 * Undecidable even if ZFC is augmented by taking the generalized continuum hypothesis as an axiom;
 * Positively answered if ZFC is augmented with the axiom of constructibility (see statements true in L).

A note on typography
Among mathematical adjectives derived from the proper name of a mathematician, the word "abelian" is rare in that it is often spelled with a lowercase a, rather than an uppercase A, the lack of capitalization being a tacit acknowledgment not only of the degree to which Abel's name has been institutionalized but also of how ubiquitous in modern mathematics are the concepts introduced by him.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Abelian group


text:
Anti-Ballistic Missile Treaty

The Anti-Ballistic Missile Treaty, also known as the ABM Treaty or ABMT, was an arms control treaty between the United States and the Soviet Union on the limitation of the anti-ballistic missile (ABM) systems used in defending areas against ballistic missile-delivered nuclear weapons. It was intended to reduce pressures to build more nuclear weapons to maintain deterrence. Under the terms of the treaty, each party was limited to two ABM complexes, each of which was to be limited to 100 anti-ballistic missiles.

Signed in 1972, it was in force for the next 30 years. In 1997, five years after the dissolution of the Soviet Union, four former Soviet republics agreed with the United States to succeed the USSR's role in the treaty, with Russia assuming all rights and obligations as the successor state of the Soviet Union. Citing risks of nuclear blackmail, the United States withdrew from the treaty in June 2002, leading to its termination.

Background
Throughout the late 1950s and into the 1960s, the United States and the Soviet Union had been developing missile systems with the ability to shoot down incoming Intercontinental ballistic missile (ICBM) warheads. During this period, the US considered the defense of the US as part of reducing the overall damage inflicted in a full nuclear exchange. As part of this defense, Canada and the US established the North American Air Defense Command (now called North American Aerospace Defense Command).

By the early 1950s, US research on the Nike Zeus missile system had developed to the point where small improvements would allow it to be used as the basis of an operational ABM system. Work started on a short-range, high-speed counterpart known as Sprint to provide defense for the ABM sites themselves. By the mid-1960s, both systems showed enough promise to start development of base selection for a limited ABM system dubbed Sentinel. In 1967, the US announced that Sentinel itself would be scaled down to the smaller and less expensive Safeguard. Soviet doctrine called for development of its own ABM system and return to strategic parity with the US. This was achieved with the operational deployment of the A-35 ABM system and its successors, which remain operational to this day.

The development of multiple independently targetable reentry vehicle (MIRV) systems allowed a single ICBM to deliver as many as ten separate warheads at a time. An ABM defense system could be overwhelmed with the sheer number of warheads. Upgrading it to counter the additional warheads would be economically unfeasible: The defenders required one rocket per incoming warhead, whereas the attackers could place 10 warheads on a single missile at a reasonable cost. To further protect against ABM systems, the Soviet MIRV missiles were equipped with decoys; R-36M heavy missiles carried as many as 40. These decoys would appear as warheads to an ABM, effectively requiring engagement of five times as many targets and rendering defense even less effective.

ABM Treaty
The United States first proposed an anti-ballistic missile treaty at the 1967 Glassboro Summit Conference during discussions between U.S. Secretary of Defense Robert McNamara and Chairman of the Council of Ministers of the Soviet Union Alexei Kosygin. McNamara argued both that ballistic missile defense could provoke an arms race, and that it might provoke a first-strike against the nation fielding the defense. Kosygin rejected this reasoning. They were trying to minimize the number of nuclear missiles in the world. Following the proposal of the Sentinel and Safeguard decisions on American ABM systems, the Strategic Arms Limitation Talks began in November 1969 (SALT I). By 1972 an agreement had been reached to limit strategic defensive systems. Each country was allowed two sites at which it could base a defensive system, one for the capital and one for ICBM silos.

The treaty was signed during the 1972 Moscow Summit on 26 May by the President of the United States, Richard Nixon and the General Secretary of the Communist Party of the Soviet Union, Leonid Brezhnev; and ratified by the U.S. Senate on 3 August 1972.

The 1974 Protocol reduced the number of sites to one per party, largely because neither country had developed a second site. The sites were Moscow for the USSR and the North Dakota Safeguard Complex for the US, which was already under construction.

Missiles limited by the treaty
The Treaty limited only ABMs capable of defending against "strategic ballistic missiles", without attempting to define "strategic". It was understood that both ICBMs and SLBMs are obviously "strategic". Neither country intended to stop the development of counter-tactical ABMs. The topic became disputable as soon as most potent counter-tactical ABMs started to be capable of shooting down SLBMs (SLBMs naturally tend to be much slower than ICBMs), nevertheless both sides continued counter-tactical ABM development.

After the SDI announcement
On 23 March 1983, Ronald Reagan announced the Strategic Defense Initiative, a research program into ballistic missile defense which would be "consistent with our obligations under the ABM Treaty". Reagan was wary of mutual deterrence with what he had recently called an "Evil Empire", and wanted to escape the traditional confines of mutual assured destruction. The project was a blow to Yuri Andropov's so-called "peace offensive". Andropov said that "It is time [Washington] stopped thinking up one option after another in search of the best way of unleashing nuclear war in the hope of winning it. To do this is not just irresponsible. It is madness".

Regardless of the opposition, Reagan gave every indication that SDI would not be used as a bargaining chip and that the United States would do all in its power to build the system. The Soviets were threatened because the Americans might have been able to make a nuclear first strike possible. In The Nuclear Predicament, Beckman claims that one of the central goals of Soviet diplomacy was to terminate SDI. A surprise attack from the Americans would destroy much of the Soviet ICBM fleet, allowing SDI to defeat a "ragged" Soviet retaliatory response. Furthermore, if the Soviets chose to enter this new arms race, they would further cripple their economy. The Soviets could not afford to ignore Reagan's new endeavor, therefore their policy at the time was to enter negotiations with the Americans. By 1987, however, the USSR withdrew its opposition, concluding the SDI posed no threat and scientifically "would never work".

SDI research went ahead, although it did not achieve the hoped-for result. SDI research was cut back following the end of Reagan's presidency, and in 1995 it was reiterated in a presidential joint statement that "missile defense systems may be deployed... [that] will not pose a realistic threat to the strategic nuclear force of the other side and will not be tested to... [create] that capability." This was reaffirmed in 1997.

Theater Missile Defense negotiations
The ABM Treaty prohibited "National Missile Defense" (NMD), but some interpreted it to allow more limited systems called "Theater Missile Defense" (TMD). This is because Article II of the treaty defined "ABM Systems" as those that "counter strategic missiles", which are typically defined as those with "intercontinental capability". Thus, TMD supporters argued, the treaty did not prohibit systems that defended against the countering of theatre ballistic missiles. The US had already developed and used such systems, including the Patriot Missile during the Gulf War.

The problem arose as TMD systems could also potentially be capable of countering strategic ballistic missiles, not just theatre ballistic missiles. The Clinton administration began negotiations with the Russians in 1993 to make amendments to the treaty. After much discussion, President Clinton and Russian President Boris Yeltsin signed addendum to the treaty on September 9, 1997. According to these new agreements, the treaty permitted missile defense systems to have a velocity up to 5 km/s as long as it had not been tested against targets traveling faster than 5 km/s.

The 1997 agreement were eventually ratified by the Russian parliament on May 4, 2000 (along with START II treaty). However, it was opposed in the U.S. Senate by some Republican senators led by Jesse Helms. As a result, Clinton never submitted the agreement to Congress, fearing that Helms would stall their ratification or defeat it outright.

After the dissolution of the USSR; United States and Russia
Although the Soviet Union ceased to exist in December 1991, in the view of the U.S. Department of State, the treaty continued in force. Russia was confirmed as the USSR's successor state in January 1992. Belarus and Ukraine were treated as successors at the ABM review conference in October 1993 and Kazakhstan was added as a successor shortly after. Belarus, Ukraine and Kazakhstan became regular participants at ABM treaty meetings known as Standing Consultative Commissions. An additional memorandum of understanding was prepared in 1997, establishing Belarus, Kazakhstan, the Russian Federation, and Ukraine as successor states to the Soviet Union, for the purposes of the treaty. The US considered only extending the obligations to these countries, and not all, as only these ones had significant ABM assets. As the ABM treaty allowed for only a single ABM deployment, the State Department deemed that only a single ABM system would be collectively permitted among Russia, Ukraine, Kazakhstan and Belarus.

In the United States, there was a debate on whether after the dissolution of the USSR, the ABM Treaty was still in effect. A month after the USSR's dissolution, President George H. W. Bush affirmed the ABM Treaty and regarded Russia as USSR's successor. Russia also accepted the ABM Treaty. Later on, President Clinton would affirm the validity of the treaty, as would President George W. Bush (before he terminated it). However, some Americans (mostly conservative Republicans ) argued that the treaty was not in effect because the USSR had no successor state. This was deemed inconsistent, as Russia did indeed inherit the USSR's obligations (including its UNSC seat, its debts, its agreements on nonproliferation etc.). Former CIA director James Woolsey argued that in order for the treaty to remain in force, both the US and Russia had to accept it, and that President Clinton could not accept it without Congressional approval. According to Michael J. Glennon, Congress acknowledged the treaty in 1996, when it passed a law restricting President Clinton's ability to modify the treaty.

United States withdrawal
On 13 December 2001, George W. Bush gave Russia notice of the United States' withdrawal from the treaty, in accordance with the clause that required six months' notice before terminating the pact—the first time in recent history that the United States has withdrawn from a major international arms treaty. This led to the eventual creation of the American Missile Defense Agency.

Supporters of the withdrawal argued that it was a necessity in order to test and build a limited National Missile Defense to protect the United States from nuclear blackmail by a rogue state. But, the withdrawal had many foreign and domestic critics, who said the construction of a missile defense system would lead to fears of a U.S. nuclear first strike, as the missile defense could blunt the retaliatory strike that would otherwise deter such a preemptive attack. John Rhinelander, a negotiator of the ABM treaty, predicted that the withdrawal would be a "fatal blow" to the Non-Proliferation Treaty and would lead to a "world without effective legal constraints on nuclear proliferation". Former U.S. Secretary of Defense William Perry also criticized the U.S. withdrawal as a very bad decision.

Russian response
Newly elected Russian president Vladimir Putin responded to the withdrawal by ordering a build-up of Russia's nuclear capabilities, designed to counterbalance U.S. capabilities, although he noted there was no immediate danger stemming from the US withdrawal.

Russia and the United States signed the Strategic Offensive Reductions Treaty in Moscow on 24 May 2002. This treaty mandates cuts in deployed strategic nuclear warheads, but without actually mandating cuts to total stockpiled warheads, and without any mechanism for enforcement.

On June 13, 2002, the US withdrew from ABM (having given notice 6 months earlier). The next day, Russia responded by declaring it would no longer abide by the START II treaty, which had not entered into force.

In interviews with Oliver Stone in 2017, Russian president Vladimir Putin said that in trying to persuade Russia to accept US withdrawal from the treaty, both Bill Clinton and George W. Bush had tried, without evidence, to convince him of an emerging nuclear threat from Iran.

On 1 March 2018, Russian president Vladimir Putin, in an address to the Federal Assembly, announced the development of a series of technologically new "super weapons" in response to U.S. withdrawal from the ABM Treaty. His statements were referred to by an anonymous US official under the Trump administration as largely boastful untruths. He said that the U.S. decision triggered him to order an increase in Russia's nuclear capabilities, designed to counterbalance U.S. ones.

In 2021, Putin cited U.S. withdrawal among his grievances against the West: "We tried to partner with the West for many years, but the partnership was not accepted, it didn't work," often citing it as one of America's great post-Cold War sins.

title: Anti-Ballistic Missile Treaty


text:
Talk:Arab–Israeli conflict

Extended-protected edit request, 30 December 2023
Greece should be removed from the infobox, since it didn't let the U.S. use its bases to support the airlift in 1973.

"During the 1973 Yom Kippur war, the U.S. arms airlift to Israel was seriously complicated by the denial of transit bases in Spain, Turkey, and Greece and overflight rights in other NATO areas."

"On October 22, 1973, MAC changed the route to fly south of Crete, to comply with a request from the Greek Government...... This zigzagging route was used because the United States could not obtain diplomatic clearance to use bases, which MAC usually used, in the United Kingdom, Spain, Italy, Greece and Turkey."

"Secondly, during the October 1973 War, Greece observed strict neutrality and refused land, air or port facilities for the airlift of American supplies to Israel. As the Greek Foreign Minister flatly stated, U.S. bases in Greece had "nothing to do with the Arab-Israeli war'."

-- 2804:29B8:5183:100C:85AA:25EF:674D:9F6D (talk) 21:58, 30 December 2023 (UTC)
 * Red information icon with gradient background.svg Not done for now: please establish a consensus for this alteration before using the template. Skitash (talk) 22:14, 30 December 2023 (UTC)

 * ✅ I checked the source given in the infobox and found that it also emphasises Greek neutrality in 1973. So there is no case for keeping it. I'll also take the opportunity to complain in general about the "Supported by" sections of these infoboxes, which often list countries whose contribution was trivial. "Portugal" should be checked too. Zerotalk 04:06, 31 December 2023 (UTC)

Greece
Greece did not support Israel in the 1973 war, it declared neutrality and didn't allow the U.S. to use its bases for the airlift, it should be removed from the infobox. -- 2804:29B8:5183:100C:D1BA:C8A:C094:DF5B (talk) 22:36, 30 December 2023 (UTC)

 * Can you provide a reliable source for this?GrayStorm(Talk|Contributions) 02:05, 13 February 2024 (UTC)

Map needs Yemen
Map needs Ansar Allah in Yemen. Irtapil (talk) 23:01, 15 January 2024 (UTC)

Extended-confirmed-protected edit request on 17 February 2024
For Belligerents on Arab side- Please add Pakistan (1948-Present) Please see the following for reference: Kumaraswamy, P. R. “Pakistan and the Arab-Israeli Conflict.” Beyond the Veil: Israel-Pakistan Relations, Institute for National Security Studies, 2000, pp. 20–29. JSTOR, http://www.jstor.org/stable/resrep08947.6. Accessed 17 Feb. 2024.

Rh272 (talk) 06:45, 17 February 2024 (UTC)
 * Red information icon with gradient background.svg Not done: The source says "Pakistan has a mixed and complex track record concerning the Arab- Israeli conflict" and "various Pakistani leaders have pursued a pragmatic approach toward Israel and its conflict with the Arabs". Skitash (talk) 11:23, 17 February 2024 (UTC)

Update info in the Template
In the Israeli side,

There are over 5,000 Israeli civilian victims. Only in 1947-2022 over 4,200 Israeli civilians were killed, not including over 800 Israeli civilians were killed in 2023. Showing less than 1,800 Israeli civilians were killed is misleading. IdanST (talk) 16:27, 1 March 2024 (UTC)

Subset
What argument are you basing this claim on? The Israeli-Palestinian conflict dates back to early 1900s, meanwhile first instance of the Arab-Israeli conflict came later around 1920. Also, Jordan and Egypt making peace with Israel clearly hasn't solved the IP conflict. Makeandtoss (talk) 13:24, 26 May 2024 (UTC)

 * Palestinians are Arabs, not Arabs are Palestinians. Israeli-Palestinian conflict is the core of this conflict. Parham wiki (talk) 04:11, 27 May 2024 (UTC)

title: Talk:Arab–Israeli conflict