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American bison

The American bison (Bison bison; : bison), also called the American buffalo or simply buffalo (not to be confused with true buffalo), is a species of bison native to North America. It is one of two extant species of bison, alongside the European bison. Its historical range circa 9000 BC is described as the great bison belt, a tract of rich grassland from Alaska to the Gulf of Mexico, east to the Atlantic Seaboard (nearly to the Atlantic tidewater in some areas), as far north as New York, south to Georgia, and according to some sources, further south to northern Florida, with sightings in North Carolina near Buffalo Ford on the Catawba River as late as 1750.

Once roaming in vast herds, the species nearly became extinct by a combination of commercial hunting and slaughter in the 19th century and introduction of bovine diseases from domestic cattle. With an estimated population of 60 million in the late 18th century, the species was culled down to just 541 animals by 1889 as part of the subjugation of the Native Americans, because the American bison was a major resource for their traditional way of life (food source, hides for clothing and shelter, and horns and bones for tools). Recovery efforts expanded in the mid-20th century, with a resurgence to roughly 31,000 wild bison as of March 2019. For many years, the population was primarily found in a few national parks and reserves. Through multiple reintroductions, the species now freely roams wild in several regions in the United States, Canada and Mexico. The American bison has also been introduced to Yakutia in Russia.

Two subspecies or ecotypes have been described: the plains bison (B. b. bison), smaller in size and with a more rounded hump, and the wood bison (B. b. athabascae)—the larger of the two and having a taller, square hump. Furthermore, the plains bison has been suggested to consist of a northern plains (B. b. montanae) and a southern plains (B. b. bison) subspecies, bringing the total to three. However, this is generally not supported. The wood bison is one of the largest wild species of extant bovid in the world, surpassed only by the Asian gaur. Among extant land animals in North America, the bison is the heaviest and the longest, and the second tallest after the moose.

Spanning back millennia, Native American tribes have had cultural and spiritual connections to the American bison. It is the national mammal of the United States.

Etymology

In American English, both buffalo and bison are considered correct terms for the American bison. However, in British English, the word buffalo is reserved for the African buffalo and water buffalo and not used for the bison.

In English usage, the term buffalo was used to refer to the American mammal as early as 1625. The word bison was applied in the 1690s.

Buffalo was applied to the American bison by Samuel de Champlain as the French word buffles in 1616 (published 1619), after seeing skins and a drawing. These were shown to him by members of the Nipissing First Nation, who said they traveled forty days (from east of Lake Huron) to trade with another nation who hunted the animals. Buffel in turn comes from Portuguese bufalo (water buffalo), which comes from Latin bufalus (an antelope, gazelle, or wild ox), from Greek boubalos. From the same Greek word boubalos we also get the Bubal hartebeest.

Bison was borrowed from French bison in the early 17th century, from Latin bison (aurochs), from a Proto-Germanic word similar to wisent and, per Etymonline, first applied to American buffalo in the 1690s.

In Plains Indian languages in general, male and female bison are distinguished, with each having a different designation rather than there being a single generic word covering both sexes. Thus: Such a distinction is not a general feature of the language (for example, Arapaho possesses gender-neutral terms for other large mammals such as elk, mule deer, etc.), and so presumably is due to the special significance of the bison in Plains Indian life and culture.
 * in Arapaho: bii (bison cow), henéécee (bison bull)
 * in Lakota: pté (bison cow), tȟatȟáŋka (bison bull)

Description
A bison has a shaggy, long, dark-brown winter coat, and a lighter-weight, lighter-brown summer coat. Male bison are significantly larger and heavier than females. Plains bison are often in the smaller range of sizes, and wood bison in the larger range. Head-rump lengths at maximum up to 3.5 m for males and 2.85 m for females long and the tail adding 30 to 95 cm. Heights at withers in the species can reach up to 186 to 201 cm for B. b. bison and B. b. athabascae respectively. Typically weights can range from 318 to 1179 kg, 460 to 988 kg with medians of 730 to 792.5 kg (B.b. bison) and 943.6 kg (B.b.athabascae) in males, and 360 to 640 kg with medians of 450 to 497.6 kg in females, although the lowest weights probably representing typical weight around the age of sexual maturity at 2 to 3 years of age.

The heaviest wild bull for B.b.bison ever recorded weighed 1270 kg while there had been bulls estimated to be 3000 lb. B.b.athabascae is significantly larger and heavier on average than B.b.bison while the number of recorded samples for the former was limited after the rediscovery of a relatively pure herd. Elk Island National Park, which has wild populations of both wood and plains bison, has recorded maximum weights for bull bison of 1186 kg (plains) and 1099 kg (wood), but noted that 3/4 of all bison over 1000 kg were wood bison. When raised in captivity and farmed for meat, the bison can grow unnaturally heavy and the largest semidomestic bison weighed 1724 kg. The heads and forequarters are massive, and both sexes have short, curved horns that can grow up to 60 cm long with 90 cm to 124 cm width, which they use in fighting for status within the herd and for defense.

Bison are herbivores, grazing on the grasses and sedges of the North American prairies. Their daily schedule involves two-hour periods of grazing, resting, and cud chewing, then moving to a new location to graze again. Sexually mature young bulls may try to start mating with cows by the age of two or three years, but if more mature bulls are present, they may not be able to compete until they reach five years of age.

For the first two months of life, calves are lighter in color than mature bison. One extremely rare condition is the white buffalo, in which the calf turns entirely white.

Evolution
Bison are members of the tribe Bovini. Genetic evidence from nuclear DNA indicates that the closest living relatives of bison are yaks, with bison being nested within the genus Bos, rendering Bos without including bison paraphyletic. While nuclear DNA indicates that the two living bison species are each other's closest living relatives, the mitochondrial DNA of European bison is more closely related to that of domestic cattle and aurochs, which is either suggested to be the result of incomplete lineage sorting or ancient introgression. Bison first appeared in Asia during the Early Pleistocene, around 2.6 million years ago. Bison only arrived in North America 195,000 to 135,000 years ago, during the late Middle Pleistocene, descending from the widespread Siberian steppe bison (Bison priscus), which had migrated through Beringia. Following its first appearance in North America, the bison rapidly differentiated into new species such as the largest of all bison, the long-horned Bison latifrons as well as Bison antiquus. The first appearance of bison in North America is considered to define the regional Rancholabrean faunal stage, due to its major impact on the ecology of the continent. Modern American bison are thought to have evolved from B. antiquus at the end of the Late Pleistocene - beginning of the Holocene, with likely intermediates between the species referred to as Bison "occidentalis".

Differences from European bison
Although they are superficially similar, the American and European bison exhibit a number of physical and behavioral differences. Adult American bison are slightly heavier on average because of their less rangy build and have shorter legs, which render them slightly shorter at the shoulder. American bison tend to graze more and browse less than their European relatives because their necks are set differently. Compared to the nose of the American bison, that of the European species is set farther forward than the forehead when the neck is in a neutral position. The body of the American bison is hairier, though its tail has less hair than that of the European bison. The horns of the European bison point forward through the plane of its face, making it more adept at fighting through the interlocking of horns in the same manner as domestic cattle, unlike the American bison, which favors charging. American bison are more easily tamed than the European and breed more readily with domestic cattle.

Crossbreeding with cattle
During the population bottleneck, after the great slaughter of American bison during the 19th century, the number of bison remaining alive in North America declined to as low as 541. During that period, a handful of ranchers gathered remnants of the existing herds to save the species from extinction. These ranchers bred some of the bison with cattle in an effort to produce "cattalo" or "beefalo". Accidental crossings were also known to occur. Generally, male domestic bulls were crossed with bison cows, producing offspring of which only the females were fertile. The crossbred animals did not demonstrate any form of hybrid vigor, so the practice was abandoned. The proportion of cattle DNA that has been measured in introgressed individuals and bison herds today is typically quite low, ranging from 0.56 to 1.8%. In the United States, many ranchers are now using DNA testing to cull the residual cattle genetics from their bison herds. The U.S. National Bison Association has adopted a code of ethics which prohibits its members from deliberately crossbreeding bison with any other species.

Range and population
Population estimates in 2010 ranged from 400,000 to 500,000, with approximately 20,500 animals in 62 conservation herds and the remainder in approximately 6,400 commercial herds. According to the IUCN, roughly 15,000 bison are considered wild, free-range bison not primarily confined by fencing.

The Nature Conservancy (TNC) has reintroduced bison to over a dozen nature preserves around the United States. In October 2016, TNC established its easternmost bison herd in the country, at Kankakee Sands nature preserve in Morocco, Newton County, Indiana. In 2014, U.S. Tribes and Canadian First Nations signed a treaty to help with the restoration of bison, the first to be signed in nearly 150 years.

Habitat and trails
American bison live in river valleys, and on prairies and plains. Typical habitat is open or semiopen grasslands, as well as sagebrush, semiarid lands, and scrublands. Some lightly wooded areas are also known historically to have supported bison. Bison also graze in hilly or mountainous areas where the slopes are not steep. Though not particularly known as high-altitude animals, bison in the Yellowstone Park bison herd are frequently found at elevations above 8000 ft, and the Henry Mountains bison herd is found on the plains around the Henry Mountains, Utah, as well as in mountain valleys of the Henry Mountains to an altitude of 10000 ft. Reintroduced plains bison in Banff National Park have been observed to roam mountainous areas, including high ridges and steep drainages, and archaeological finds indicate that some bison historically may have spent their lives within mountains while others may have migrated in and out of mountains.

Those in Yukon, Canada, typically summer in alpine plateaus above treeline. The first thoroughfares of North America, except for the time-obliterated paths of mastodon or muskox and the routes of the mound builders, were the traces made by bison and deer in seasonal migration and between feeding grounds and salt licks. Many of these routes, hammered by countless hoofs instinctively following watersheds and the crests of ridges in avoidance of lower places' summer muck and winter snowdrifts, were followed by the aboriginal North Americans as courses to hunting grounds and as warriors' paths. They were invaluable to explorers and were adopted by pioneers.

Bison traces were characteristically north and south, but several key east–west trails were used later as railways. Some of these include the Cumberland Gap through the Blue Ridge Mountains to upper Kentucky. A heavily used trace crossed the Ohio River at the Falls of the Ohio and ran west, crossing the Wabash River near Vincennes, Indiana. In Senator Thomas Hart Benton's phrase saluting these sagacious path-makers, the bison paved the way for the railroads to the Pacific.

Mexico
The southern extent of the historic range of the American bison includes northern Mexico and adjoining areas in the United States as documented by archeological records and historical accounts from Mexican archives from 700 CE to the 19th century. The Janos-Hidalgo bison herd has ranged between Chihuahua, Mexico, and New Mexico, United States, since at least the 1920s. The persistence of this herd suggests that habitat for bison is suitable in northern Mexico. In 2009, genetically pure bison were reintroduced to the Janos Biosphere Reserve in northern Chihuahua adding to the Mexican bison population. In 2020, the second herd was formed in Maderas del Carmen. A private reserve named Jagüey de Ferniza has kept bisons since before the above-mentioned reintroductions in Coahuila.

Introductions to Siberia
Since 2006, an outherd of wood bison sent from Alberta's Elk Island National Park was established in Yakutia, Russia  as a practice of pleistocene rewilding; wood bison are the most similar to the extinct steppe bison species (Bison priscus). The bison are adapting well to the cold climate, and Yakutia's Red List officially registered the species in 2019; a second herd was formed in 2020.

In Pleistocene Park, there are also 24 plains bison as wood bison could not be acquired.

Behavior and ecology
Bison are migratory and herd migrations can be directional as well as altitudinal in some areas. Bison have usual daily movements between foraging sites during the summer. In the Hayden Valley, Wyoming, bison have been recorded traveling, on average, 2 mi per day. The summer ranges of bison appear to be influenced by seasonal vegetation changes, interspersion and size of foraging sites, the rut, and the number of biting insects. The size of preserve and availability of water may also be a factor. Bison are largely grazers, eating primarily grasses and sedges. On shortgrass pasture, bison predominately consume warm-season grasses. On mixed prairie, cool-season grasses, including some sedges, apparently compose 79–96% of their diet. In montane and northern areas, sedges are selected throughout the year. Bison also drink water or consume snow on a daily basis.

Social behavior and reproduction
Female bison live in maternal herds which include other females and their offspring. Male offspring leave their maternal herd when around three years old and either live alone or join other males in bachelor herds. Male and female herds usually do not mingle until the breeding season, which can occur from July through September. However, female herds may also contain a few older males. During the breeding season, dominant bulls maintain a small harem of females for mating. Individual bulls "tend" cows until allowed to mate, by following them around and chasing away rival males. The tending bull shields the female's vision with his body so she will not see any other challenging males. A challenging bull may bellow or roar to get a female's attention and the tending bull has to bellow/roar back. The most dominant bulls mate in the first 2–3 weeks of the season. More subordinate bulls mate with any remaining estrous cow that has not mated yet. Male bison play no part in raising the young.

Bison herds have dominance hierarchies that exist for both males and females. A bison's dominance is related to its birth date. Bison born earlier in the breeding season are more likely to be larger and more dominant as adults. Thus, bison are able to pass on their dominance to their offspring as dominant bison breed earlier in the season. In addition to dominance, the older bison of a generation also have a higher fertility rate than the younger ones.

Bison mate in August and September; gestation is 285 days. A single reddish-brown calf nurses until the next calf is born. If the cow is not pregnant, a calf will nurse for 18 months. Cows nurse their calves for at least 7 or 8 months, but most calves seem to be weaned before the end of their first year. At three years of age, bison cows are mature enough to produce a calf. The birthing period for bison in boreal biomes is protracted compared to that of other northern ungulates, such as moose and caribou.

Bison have a life expectancy around 15 years in the wild and up to 25 years in captivity. However, males and females from a hunted population also subject to wolf predation in northern Canada have been reported to live to 22 and 25 years of age, respectively.

Bison have been observed to display homosexual behaviors, males much more so than females. In the case of males, it is unlikely to be related to dominance, but rather to social bonding or gaining sexual experience.

Horning
Bison mate in late spring and summer in more open plain areas. During fall and winter, bison tend to gather in more wooded areas. During this time, bison partake in horning behaviors. They rub their horns against trees, young saplings, and even utility poles. Aromatic trees like cedars and pine seem to be preferred. Horning appears to be associated with insect defense, as it occurs most often in the fall when the insect population is at its highest. Cedar and pines emit an aroma after bison horn them and this seems to be used as a deterrent for insects.

Wallowing behavior
A bison wallow is a shallow depression in the soil, which bison use either wet or dry. Bison roll in these depressions, covering themselves with dust or mud. Past and current hypotheses to explain the purpose of wallowing include grooming associated with shedding, male-male interaction (typically rutting), social behavior for group cohesion, play, relief from skin irritation due to biting insects, reduction of ectoparasite (tick and lice) load, and thermoregulation. Bison wallowing has important ecosystem engineering effects and enhances plant and animal diversity on prairies.

Predation
While often secure from predation because of their size and strength, in some areas, vulnerable individuals are regularly preyed upon by wolves. Wolf predation typically peaks in late winter, when elk migrates south and bison are distressed with heavy snows and shortages of food sources, with attacks usually being concentrated on weakened and injured cows and calves. Wolves more actively target herds with calves than those without. The length of a predation episode varies, ranging from a few minutes to over nine hours. Bison display five apparent defense strategies in protecting calves from wolves: running to a cow; running to a herd; running to the nearest bull; running in the front or center of a stampeding herd; entering water bodies, such as lakes or rivers. When fleeing wolves in open areas, cows with young calves take the lead, while bulls take to the rear of the herds to guard the cows' escape. Bison typically ignore wolves not displaying hunting behavior. Wolf packs specializing in bison tend to have more males because their larger size than females allows them to wrestle prey to the ground more effectively. Healthy, mature bulls in herds rarely fall prey.

Grizzly bears are known to feed on carcass and may steal wolves' kills. While grizzlies can also pose a threat to calves and sometimes old, injured, or sick adult bison, direct killing of non-calves is rare even when targeting lone and injured young individuals; attacking healthy bison is risky for bears, who can be killed instead.

Dangers to humans
Bison are among the most dangerous animals encountered by visitors to the various North American national parks and will attack humans if provoked. They appear slow because of their lethargic movements but can easily outrun humans; bison have been observed running as fast as 40 to 45 mph. Bison may approach people for curiosity. Close encounters, including to touch the animals, can be dangerous, and gunshots do not startle them. Between 1980 and 1999, more than three times as many people in Yellowstone National Park were injured by bison than by bears. During this period, bison charged and injured 79 people, with injuries ranging from goring puncture wounds and broken bones to bruises and abrasions. Bears injured 24 people during the same time. Three people died from the injuries inflicted—one person by bison in 1983, and two people by bears in 1984 and 1986.

Genetics
A major problem that bison face today is a lack of genetic diversity due to the population bottleneck the species experienced during its near-extinction event. Another genetic issue is the entry of genes from domestic cattle into the bison population, through hybridization.

Officially, the "American buffalo" is classified by the United States government as a type of cattle, and the government allows private herds to be managed as such. This is a reflection of the characteristics that bison share with cattle. Though the American bison is a separate species and usually regarded as being in a separate genus from domestic cattle (Bos taurus), they have a lot of genetic compatibility with cattle. American bison can interbreed with cattle, although only the female offspring are fertile in the first generation. These female hybrids can be bred back to either bison or domestic bulls, resulting in either 1/4 or 3/4 bison young. Female offspring from this cross are also fertile, but males are not reliably fertile unless they are either $7/8$ bison or $7/8$ domestic. Moreover, when they do interbreed, crossbreed animals in the first generation tend to look very much like purebred bison, so appearance is completely unreliable as a means of determining what is a purebred bison and what is a crossbred cow. Many ranchers have deliberately crossbred their cattle with bison, and some natural hybridization could be expected in areas where cattle and bison occur in the same range. Since cattle and bison eat similar food and tolerate similar conditions, they have often been in the same range together in the past, and opportunity for crossbreeding may sometimes have been common.

In recent decades, tests were developed to determine the source of mitochondrial DNA in cattle and bison, and most private "buffalo" herds were actually crossbred with cattle, and even most state and federal buffalo herds had some cattle DNA. With the advent of nuclear microsatellite DNA testing, the number of herds known to contain cattle genes has increased. As of 2011, though about 500,000 bison existed on private ranches and in public herds, perhaps only 15,000 to 25,000 of these bison were pure and not actually bison-cattle hybrids. DNA from domestic cattle (Bos taurus) has been found in almost all examined bison herds.

Significant public bison herds that do not appear to have hybridized domestic cattle genes are the Yellowstone Park bison herd, the Henry Mountains bison herd, which was started with bison taken from Yellowstone Park, the Wind Cave bison herd, and the Wood Buffalo National Park bison herd and subsidiary herds started from it, in Canada.

A landmark study of bison genetics performed by James Derr of Texas A&M University corroborated this. The Derr study was undertaken in an attempt to determine what genetic problems bison might face as they repopulate former areas, and it noted that bison seem to be adapting successfully, despite their apparent genetic bottleneck. One possible explanation for this might be the small amount of domestic cattle genes that are now in most bison populations, though this is not the only possible explanation for bison success.

In the study, cattle genes were also found in small amounts throughout most national, state, and private herds. "The hybridization experiments conducted by some of the owners of the five foundation herds of the late 1800s, have left a legacy of a small amount of cattle genetics in many of our existing bison herds," said Derr. "All of the state owned bison herds tested (except for possibly one) contain animals with domestic cattle mtDNA."

It appears that the one state herd that had no cattle genes was the Henry Mountains bison herd; the Henry Mountain herd was started initially with transplanted animals from Yellowstone Park. However, the extension of this herd into the Book Cliffs of central Utah involved mixing the founders with additional bison from another source, so it is not known if the Book Cliffs extension of the herd is also free of cattle hybridization.

A separate study by Wilson and Strobeck, published in Genome, was done to define the relationships between different herds of bison in the United States and Canada, and to determine whether the bison at Wood Buffalo National Park in Canada and the Yellowstone Park bison herd were possibly separate subspecies. The Wood Buffalo Park bison were determined to actually be crossbreeds between plains and wood bison, but their predominant genetic makeup was that of the expected "wood buffalo". However, the Yellowstone Park bison herd was pure plains bison, and not any of the other previously suggested subspecies. Another finding was that the bison in the Antelope Island herd in Utah appeared to be more distantly related to other plains bison in general than any other plains bison group that was tested, though this might be due to genetic drift caused by the small size of only 12 individuals in the founder population. A side finding of this was that the Antelope Island bison herd appears to be most closely related to the Wood Buffalo National Park bison herd, though the Antelope Island bison are actually plains bison.

In order to bolster the genetic diversity of the American bison, the National Park Service alongside the Department of the Interior announced on May 7, 2020, the 2020 Bison Conservation Initiative. This initiative focuses on maintaining the genetic diversity of the metapopulation rather than individual herds. Small populations of bison are at considerably larger risk due to their decreased gene pool and are susceptible to catastrophic events more so than larger herds. The 2020 Bison Conservation Initiative aims to translocate up to three bison every five to ten years between the Department of the Interior's herds. Specific smaller herds will require a more intense management plan. Translocated bison will also be screened for any health defects such as infection of brucellosis bacteria as to not put the larger herd at risk.

Population bottlenecking and near extinction
Because of the mass slaughtering of bison during the 1870s, the plains bison population went through a population bottleneck from an estimated 60 million individuals–an estimation based on an observation made by Colonel R.I. Dodge along the Arkansas River in Kansas in 1871–to a founding population of around 100 individuals, split into six herds, five of which were managed by private ranchers and one managed by the New York Zoological Park (now the Bronx Zoo). Additionally, a wild herd consisting of 25 individuals in Yellowstone National Park survived the bottleneck.

Each of the privately ranched herds had an initial effective population size (Ne) of an estimated 5 to 7 individuals, for a total combined effective population size of between 30 and 50 individuals, from which all of the modern plains bison descend. While these herds have remained mostly isolated, some more than others, there has been some interbreeding between the herds over the past 150 years.

The conservation efforts and copious amounts of data taken on American bison populations allow for American bison to serve as a useful study case of population bottlenecking and its effects. This is especially true of the Texas State Bison Herd, which underwent very extreme genetic bottlenecking, with a founding population of only 5 individuals.

Texas State Bison Herd
The Texas State Bison Herd (TSBH), also known as the Goodnight herd, was established by Charles Goodnight in the mid-1880s with five wild-caught calves. In 1887, the herd consisted of 13 individuals; in 1910, the population consisted of 125 individuals; and in the 1920s, the population ranged from 200 to 250 individuals. In 1929, Goodnight died and the herd switched hands multiple times, leaving the population of the herd unknown from 1930 until the herd was donated to the State of Texas in 1997, with a population of 36 individuals, solely descended from the original five calves. By 2002, the population of the TSBH consisted of 40 individuals and had concerningly low birth rates and high rates of calf mortality. This led to extra attention being given to this herd by conservationists who then performed significant amounts of genetic testing.

Goodnight was an advocate for the hybridization of bison with cattle, in the hopes of creating a stronger and healthier breed. When the herd was donated to the State of Texas, genetic testing revealed that 6 out of 36 individuals still carried cattle mitochondrial DNA.

Researchers found that the average number of alleles per locus and the heterozygosity levels (a measure of genetic diversity, where high heterozygosity is representative of high genetic diversity) for the TSBH were significantly lower than that of the Yellowstone National Park bison population and the Theodore Roosevelt National Park bison population. Additionally, of the 54 nuclear microsatellites that were examined, the TSBH had 8 monomorphic loci (i.e., each loci had only one allele), whereas in both the Yellowstone and Theodore Roosevelt herds there was only one monomorphic locus, indicating a much lower level of genetic diversity in the TSBH. The Yellowstone herd had an average number of alleles per locus of 4.75, the Theodore Roosevelt National Park herd had an average of 4.15 alleles per locus, but the TSBH only had an average of 2.54 alleles per locus, statistically significantly lower than the others. The heterozygosity level of the Yellowstone, Theodore Roosevelt, and TSBH populations were 0.63, 0.57, and 0.38 respectively, with the TSBH again having a statistically significantly lower value. This low genetic diversity found in TSBH is likely due to the critically low starting population, several additional bottlenecks throughout the herd's history–leading to inbreeding depression–, and a continuously low population allowing for genetic drift to have a large effect. Before any addition of new individuals, the rate of loss of genetic diversity was estimated to be between 30 and 40% over the proceeding 50 years.

The inbreeding depression resulting from the multiple extreme population bottlenecks in the TSBH led to a coefficient of inbreeding of 0.367, equal to the level of inbreeding that results from two generations of full-siblings mating.

The Texas State Bison Herd is also a useful example of the deleterious effects of extreme population bottlenecking, with an average natality rate of 0.376 offspring per female and a 1st-year mortality rate of 52.6% from 1997 to 2002, compared to an average natality rate of 0.560 offspring per female and a 1st-year mortality rate of 4.2% for the other bison herds.

Additionally, if it were not for the intervention of conservationists, the Texas State Bison Herd would have most likely gone extinct, as the population bottleneck would have proven to be too severe. Multiple population models based on the genetics of the TSBH in the early 2000s predicted a 99% chance of extinction of the TSBH in less than 50 years, with an estimation in 2004 giving the TSBH a 99% chance of extinction in 41 years without the introduction of any outside individuals (Halbert et al. 2004). Importantly for conservation, another simulation predicted that the addition of multiple (3–9) outside male bison into the herd would increase genetic diversity enough to give the herd a 100% chance of surviving for another 100 years.

Conservation efforts have led the current TSBH population to be at the carrying capacity of their habitat, at around 300 individuals.

Yellowstone National Park Bison Herd
The Yellowstone National Park Bison herd started with only 25 individuals, and there was evidence of two population bottlenecking events from 1896 to 1912, with a population ranging between 25 and 50 individuals during this time. In 1902, 18 female and 3 male bison from outside herds–the Pablo-Allard herd and Goodnight (TSBH) herds respectively–were introduced to the Yellowstone herd. After the addition of those individuals, the effective population size is estimated to have been Ne=7.2 individuals. The Yellowstone herd was kept completely isolated from 1902 to around 1920, and these previously mentioned founders contributed between 60 and 70% of the genetics of the current bison population at Yellowstone.

Similar to the Texas State Bison Herd, the introduction of new individuals into the population in 1902 likely was the savior of this herd, which now numbers around 5,900 individuals as of summer 2022.

Population recovery
From the late 19th century onwards, the bison population gradually rose from 325 in 1884 to 500,000 in 2017, as a result of careful preservation and a general population boom. Although they are no longer classified as endangered, there are still conservation efforts in order to prevent population crashes down the line.

Hunting
Buffalo hunting, i.e. hunting of the American bison, was an activity fundamental to the Indigenous peoples of the Great Plains, providing more than 150 uses for all parts of the animal, including being a major food source, hides for clothing and shelter, bones and horns as tools as well as ceremonial and adornment uses. Bison hunting was later adopted by American professional hunters, as well as by the U.S. government, in an effort to sabotage the central resource of some American Indian Nations during the later portions of the American Indian Wars, leading to the near-extinction of the species around 1890. For many tribes the buffalo was an integral part of life—something guaranteed to them by the Creator. In fact, for some Plains indigenous peoples, bison are known as the first people. The concept of species extinction was foreign to many tribes.

Thus, when the U.S. government began to massacre the buffalo, it was particularly harrowing to the Indigenous people. As Crow chief Plenty Coups described it: "When the buffalo went away the hearts of my people fell to the ground, and they could not lift them up again. After this nothing happened. There was little singing anywhere." Spiritual loss was rampant; bison were an integral part of traditional tribal societies, and they would frequently take part in ceremonies for each bison they killed to honor its sacrifice. In order to boost morale during this time, Sioux and other tribes took part in the Ghost Dance, which consisted of hundreds of people dancing until 100 persons were lying unconscious.

Today, many conservation measures have been taken by Native Americans, with the Inter Tribal Bison Council being one of the most significant. It was formed in 1990, composed of 56 tribes in 19 states. These tribes represent a collective herd of more than 15,000 bison and focus on reestablishing herds on tribal lands in order to promote culture, revitalize spiritual solidarity, and restore the ecosystem. Some Inter Tribal Bison Council members argue that the bison's economic value is one of the main factors driving its resurgence. Bison serve as a low cost substitute for cattle, and can withstand the winters in the Plains region far easier than cattle.

As livestock
Bison are increasingly raised for meat, hide, wool, and dairy products. The majority of American bison in the world are raised for human consumption or fur clothing. Bison meat is generally considered to taste very similar to beef, but is lower in fat and cholesterol, yet higher in protein than beef, which has led to the development of beefalo, a fertile hybrid of bison and domestic cattle. In 2005, about 35,000 bison were processed for meat in the U.S., with the National Bison Association and USDA providing a "Certified American Buffalo" program with birth-to-consumer tracking of bison via RFID ear tags. A market even exists for kosher bison meat; these bison are slaughtered at one of the few kosher mammal slaughterhouses in the U.S., and the meat is then distributed nationwide.

Bison are found in publicly and privately held herds. Custer State Park in South Dakota is home to 1,500 bison, one of the largest publicly held herds in the world, but some question the genetic purity of the animals. Wildlife officials believe that free roaming herds with minimal cattle introgression on public lands in North America can be found only in: the Yellowstone Park bison herd; the Henry Mountains bison herd at the Book Cliffs and Henry Mountains in Utah; at Wind Cave National Park in South Dakota; Fort Peck Indian Reservation in Montana; Mackenzie Bison Sanctuary in the Northwest Territories; Elk Island National Park and Wood Buffalo National Park in Alberta; Grasslands National Park and Prince Albert National Park in Saskatchewan. Another population, the Antelope Island bison herd on Antelope Island in Utah, consisting of 550 to 700 bison, is also one of the largest and oldest public herds in the United States, but the bison in that herd are considered to be only semifree roaming, since they are confined to the Antelope Island. In addition, recent genetic studies indicate that, like most bison herds, the Antelope Island bison herd has a small number of genes from domestic cattle. In 2002, the United States government donated some bison calves from South Dakota and Colorado to the Mexican government. Their descendants live in the Mexican nature reserves El Uno Ranch at Janos and Santa Elena Canyon, Chihuahua, and Boquillas del Carmen, Coahuila, located near the southern banks of the Rio Grande, and around the grassland state line with Texas and New Mexico.

Recent genetic studies of privately owned herds of bison show that many of them include animals with genes from domestic cattle. For example, the herd on Santa Catalina Island, California, isolated since 1924 after being brought there for a movie shoot, were found to have cattle introgression. As few as 12,000 to 15,000 pure bison are estimated to remain in the world. The numbers are uncertain because the tests used to date—mitochondrial DNA analysis—indicate only if the maternal line (back from mother to mother) ever included domesticated bovines, thus say nothing about possible male input in the process. Most hybrids were found to look exactly like purebred bison; therefore, appearance is not a good indicator of genetics.

The size of the Canadian domesticated herd (genetic questions aside) grew dramatically through the 1990s and 2000s. The 2006 Census of Agriculture reported the Canadian herd at 195,728 head, a 34.9% increase since 2001. Of this total, over 95% were located in Western Canada, and less than 5% in Eastern Canada. Alberta was the province with the largest herd, accounting for 49.7% of the herd and 45.8% of the farms. The next-largest herds were in Saskatchewan (23.9%), Manitoba (10%), and British Columbia (6%). The main producing regions were in the northern parts of the Canadian prairies, specifically in the parkland belt, with the Peace River region (shared between Alberta and British Columbia) being the most important cluster, accounting for 14.4% of the national herd. Canada also exports bison meat, totaling 2075253 kg in 2006.

A proposal known as Buffalo Commons has been suggested by a handful of academics and policymakers to restore large parts of the drier portion of the Great Plains to native prairie grazed by bison. Proponents argue that current agricultural use of the shortgrass prairie is not sustainable, pointing to periodic disasters, including the Dust Bowl, and continuing significant human population loss over the last 60 years. However, this plan is opposed by some who live in the areas in question.

Domestication
Despite being the closest relatives of domestic cattle native to North America, bison were never domesticated by Native Americans. Later attempts of domestication by Europeans prior to the 20th century met with limited success. Bison were described as having a "wild and ungovernable temper"; they can jump close to 6 ft vertically, and run 35 - 45 mph when agitated. This agility and speed, combined with their great size and weight, makes bison herds difficult to confine, as they can easily escape or destroy most fencing systems, including most razor wire. The most successful systems involve large, 20 ft fences made from welded steel I beams sunk at least 6 ft into concrete. These fencing systems, while expensive, require very little maintenance. Furthermore, making the fence sections overlap so the grassy areas beyond are not visible prevents the bison from trying to get to new range.

Native Americans
Among many Native American tribes, especially the Plains Indians, the bison is considered a sacred animal and religious symbol. According to University of Montana anthropology and Native American studies professor S. Neyooxet Greymorning, "The creation stories of where buffalo came from put them in a very spiritual place among many tribes. The buffalo crossed many different areas and functions, and it was utilized in many ways. It was used in ceremonies, as well as to make tipi covers that provided homes for people, utensils, shields, weapons and parts were used for sewing with the sinew." The Sioux consider the birth of a white buffalo to be the return of White Buffalo Calf Woman, their primary cultural prophet and the bringer of their "Seven Sacred Rites". Among the Mandan and Hidatsa, the White Buffalo Cow Society was the most sacred of societies for women.

North America
The American bison is often used in North America in official seals, flags, and logos. In 2016, the American bison became the national mammal of the United States. The bison is a popular symbol in the Great Plains states: Kansas, Oklahoma, and Wyoming have adopted the animal as their official state mammal, and many sports teams have chosen the bison as their mascot. In Canada, the bison is the official animal of the province of Manitoba and appears on the Manitoba flag. It is also used in the official coat of arms of the Royal Canadian Mounted Police.

Several American coins feature the bison, most famously on the reverse side of the "buffalo nickel" from 1913 to 1938. In 2005, the United States Mint coined a nickel with a new depiction of the bison as part of its "Westward Journey" series. The Kansas and North Dakota state quarters, part of the "50 State Quarter" series, each feature bison. The Kansas state quarter has only the bison and does not feature any writing, while the North Dakota state quarter has two bison. The Montana state quarter prominently features a bison skull over a landscape. The Yellowstone National Park quarter also features a bison standing next to a geyser.

Other institutions which have adopted the bison as a symbol or mascot include: • U.S. Department of the Interior

• Bethany College (West Virginia)

• Bucknell University and its athletic program, the Bucknell Bison

• Buffalo, New York

• Buffalo Bills

• Buffalo Bisons

• Buffalo Gap High School

• Buffalo Grove High School

• Buffalo Sabres

• University of Colorado and its athletic program, the Colorado Buffaloes

• Gallaudet University

• Harding University and its athletic program, the Harding Bisons

• Howard University and its athletic program, the Howard Bison

• Seal of the State of Indiana

• Lipscomb University and its athletic program, the Lipscomb Bisons

• Coat of arms of Manitoba

• Flag of Manitoba

• University of Manitoba and its athletic program, the Manitoba Bisons

• Marshall University and its athletic program, the Marshall Thundering Herd

• Milligan University

• Independence Party of Minnesota

• Ralph Nader (mascot for his 2008 campaign for president)

• Nichols College

• North Dakota State University and its athletic program, the North Dakota State Bison

• Oklahoma Baptist University and its athletic program, the Oklahoma Baptist Bison

• Point Park University

• Royal Canadian Mounted Police

• Rumble the Bison (the official mascot of the Oklahoma City Thunder)

• Smoky Hill High School

• Southwestern Law School

• Tooele High School (Utah)

• Utah Tech University and its athletic program, the Utah Tech Trailblazers

• CFB Wainwright

• West Texas A&M University and its athletic program, the West Texas A&M Buffaloes

• Regional Municipality of Wood Buffalo

• Bloomington Bison

title: American bison


text:
Alternating group

In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of $n$ elements is called the alternating group of degree $n$, or the alternating group on $n$ letters and denoted by $An$ or $Alt(n).$

Basic properties
For n > 1, the group An is the commutator subgroup of the symmetric group Sn with index 2 and has therefore n!/2 elements. It is the kernel of the signature group homomorphism sgn : Sn → $\{1, −1\}$ explained under symmetric group.

The group An is abelian if and only if n ≤ 3 and simple if and only if n = 3 or n ≥ 5. A5 is the smallest non-abelian simple group, having order 60, and the smallest non-solvable group.

The group A4 has the Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions $\{, (12)(34), (13)(24), (14)(23) \}$, that is the kernel of the surjection of A4 onto A3 ≅ Z3. We have the exact sequence V → A4 → A3 = Z3. In Galois theory, this map, or rather the corresponding map S4 → S3, corresponds to associating the Lagrange resolvent cubic to a quartic, which allows the quartic polynomial to be solved by radicals, as established by Lodovico Ferrari.

Conjugacy classes
As in the symmetric group, any two elements of An that are conjugate by an element of An must have the same cycle shape. The converse is not necessarily true, however. If the cycle shape consists only of cycles of odd length with no two cycles the same length, where cycles of length one are included in the cycle type, then there are exactly two conjugacy classes for this cycle shape.

Examples:
 * The two permutations (123) and (132) are not conjugates in A3, although they have the same cycle shape, and are therefore conjugate in S3.
 * The permutation (123)(45678) is not conjugate to its inverse (132)(48765) in A8, although the two permutations have the same cycle shape, so they are conjugate in S8.

Relation with symmetric group

 * See Symmetric group.

As finite symmetric groups are the groups of all permutations of a set with finite elements, and the alternating groups are groups of even permutations, alternating groups are subgroups of finite symmetric groups.

Generators and relations
For n ≥ 3, An is generated by 3-cycles, since 3-cycles can be obtained by combining pairs of transpositions. This generating set is often used to prove that An is simple for n ≥ 5.

Automorphism group
For n > 3, except for n = 6, the automorphism group of An is the symmetric group Sn, with inner automorphism group An and outer automorphism group Z2; the outer automorphism comes from conjugation by an odd permutation.

For n = 1 and 2, the automorphism group is trivial. For n = 3 the automorphism group is Z2, with trivial inner automorphism group and outer automorphism group Z2.

The outer automorphism group of A6 is the Klein four-group V = Z2 × Z2, and is related to the outer automorphism of S6. The extra outer automorphism in A6 swaps the 3-cycles (like (123)) with elements of shape 32 (like (123)(456)).

Exceptional isomorphisms
There are some exceptional isomorphisms between some of the small alternating groups and small groups of Lie type, particularly projective special linear groups. These are:
 * A4 is isomorphic to PSL2(3) and the symmetry group of chiral tetrahedral symmetry.
 * A5 is isomorphic to PSL2(4), PSL2(5), and the symmetry group of chiral icosahedral symmetry. (See for an indirect isomorphism of PSL2(F5) → A5 using a classification of simple groups of order 60, and here for a direct proof).
 * A6 is isomorphic to PSL2(9) and PSp4(2)'.
 * A8 is isomorphic to PSL4(2).

More obviously, A3 is isomorphic to the cyclic group Z3, and A0, A1, and A2 are isomorphic to the trivial group (which is also SL1(q) = PSL1(q) for any q).

Example A5 as a subgroup of 3-space rotations

A5 is the group of isometries of a dodecahedron in 3-space, so there is a representation A5 → SO3(R).

In this picture the vertices of the polyhedra represent the elements of the group, with the center of the sphere representing the identity element. Each vertex represents a rotation about the axis pointing from the center to that vertex, by an angle equal to the distance from the origin, in radians. Vertices in the same polyhedron are in the same conjugacy class. Since the conjugacy class equation for A5 is 1 + 12 + 12 + 15 + 20 = 60, we obtain four distinct (nontrivial) polyhedra.

The vertices of each polyhedron are in bijective correspondence with the elements of its conjugacy class, with the exception of the conjugacy class of (2,2)-cycles, which is represented by an icosidodecahedron on the outer surface, with its antipodal vertices identified with each other. The reason for this redundancy is that the corresponding rotations are by π radians, and so can be represented by a vector of length π in either of two directions. Thus the class of (2,2)-cycles contains 15 elements, while the icosidodecahedron has 30 vertices.

The two conjugacy classes of twelve 5-cycles in A5 are represented by two icosahedra, of radii 2π/5 and 4π/5, respectively. The nontrivial outer automorphism in Out(A5) ≃ Z2 interchanges these two classes and the corresponding icosahedra.

Example: the 15 puzzle
It can be proved that the 15 puzzle, a famous example of the sliding puzzle, can be represented by the alternating group A15, because the combinations of the 15 puzzle can be generated by 3-cycles. In fact, any 2k − 1 sliding puzzle with square tiles of equal size can be represented by A2k−1.

Subgroups
A4 is the smallest group demonstrating that the converse of Lagrange's theorem is not true in general: given a finite group G and a divisor d of $|G|$, there does not necessarily exist a subgroup of G with order d: the group G = A4, of order 12, has no subgroup of order 6. A subgroup of three elements (generated by a cyclic rotation of three objects) with any distinct nontrivial element generates the whole group.

For all n > 4, An has no nontrivial (that is, proper) normal subgroups. Thus, An is a simple group for all n > 4. A5 is the smallest non-solvable group.

Group homology
The group homology of the alternating groups exhibits stabilization, as in stable homotopy theory: for sufficiently large n, it is constant. However, there are some low-dimensional exceptional homology. Note that the homology of the symmetric group exhibits similar stabilization, but without the low-dimensional exceptions (additional homology elements).

H1: Abelianization
The first homology group coincides with abelianization, and (since An is perfect, except for the cited exceptions) is thus:
 * H1(An, Z) = Z1 for n = 0, 1, 2;
 * H1(A3, Z) = A$ab 3$ = A3 = Z3;
 * H1(A4, Z) = A$ab 4$ = Z3;
 * H1(An, Z) = Z1 for n ≥ 5.

This is easily seen directly, as follows. An is generated by 3-cycles – so the only non-trivial abelianization maps are An → Z3, since order-3 elements must map to order-3 elements – and for n ≥ 5 all 3-cycles are conjugate, so they must map to the same element in the abelianization, since conjugation is trivial in abelian groups. Thus a 3-cycle like (123) must map to the same element as its inverse (321), but thus must map to the identity, as it must then have order dividing 2 and 3, so the abelianization is trivial.

For n < 3, An is trivial, and thus has trivial abelianization. For A3 and A4 one can compute the abelianization directly, noting that the 3-cycles form two conjugacy classes (rather than all being conjugate) and there are non-trivial maps A3 ↠ Z3 (in fact an isomorphism) and A4 ↠ Z3.

H2: Schur multipliers
The Schur multipliers of the alternating groups An (in the case where n is at least 5) are the cyclic groups of order 2, except in the case where n is either 6 or 7, in which case there is also a triple cover. In these cases, then, the Schur multiplier is (the cyclic group) of order 6. These were first computed in.

 * H2(An, Z) = Z1 for n = 1, 2, 3;
 * H2(An, Z) = Z2 for n = 4, 5;
 * H2(An, Z) = Z6 for n = 6, 7;
 * H2(An, Z) = Z2 for n ≥ 8.

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

title: Alternating group


text:
Parity of a permutation

In mathematics, when X is a finite set with at least two elements, the permutations of X (i.e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation $$\sigma$$ of X can be defined as the parity of the number of inversions for σ, i.e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y).

The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi-Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non-bijective maps.

The sign of a permutation can be explicitly expressed as

where N(σ) is the number of inversions in σ.

Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as

where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well-defined.

Example
Consider the permutation σ of the set $\{1, 2, 3, 4, 5\}$ defined by $$\sigma(1) = 3,$$ $$\sigma(2) = 4,$$ $$\sigma(3) = 5,$$ $$\sigma(4) = 2,$$ and $$\sigma(5) = 1.$$ In one-line notation, this permutation is denoted 34521. It can be obtained from the identity permutation 12345 by three transpositions: first exchange the numbers 2 and 4, then exchange 3 and 5, and finally exchange 1 and 3. This shows that the given permutation σ is odd. Following the method of the cycle notation article, this could be written, composing from right to left, as
 * $$\sigma=\begin{pmatrix}1&2&3&4&5\\

3&4&5&2&1\end{pmatrix} = \begin{pmatrix}1&3&5\end{pmatrix} \begin{pmatrix}2&4\end{pmatrix} = \begin{pmatrix}1&3\end{pmatrix} \begin{pmatrix}3&5\end{pmatrix} \begin{pmatrix}2&4\end{pmatrix} .$$ There are many other ways of writing σ as a composition of transpositions, for instance

but it is impossible to write it as a product of an even number of transpositions.

Properties
The identity permutation is an even permutation. An even permutation can be obtained as the composition of an even number (and only an even number) of exchanges (called transpositions) of two elements, while an odd permutation can be obtained by (only) an odd number of transpositions.

The following rules follow directly from the corresponding rules about addition of integers: From these it follows that
 * the composition of two even permutations is even
 * the composition of two odd permutations is even
 * the composition of an odd and an even permutation is odd
 * the inverse of every even permutation is even
 * the inverse of every odd permutation is odd

Considering the symmetric group Sn of all permutations of the set {1, ..., n}, we can conclude that the map

that assigns to every permutation its signature is a group homomorphism.

Furthermore, we see that the even permutations form a subgroup of Sn. This is the alternating group on n letters, denoted by An. It is the kernel of the homomorphism sgn. The odd permutations cannot form a subgroup, since the composite of two odd permutations is even, but they form a coset of An (in Sn).

If n > 1, then there are just as many even permutations in Sn as there are odd ones; consequently, An contains n!/2 permutations. (The reason is that if σ is even then (1  2)σ is odd, and if σ is odd then (1  2)σ is even, and these two maps are inverse to each other.)

A cycle is even if and only if its length is odd. This follows from formulas like
 * $$(a\ b\ c\ d\ e)=(d\ e)(c\ e)(b\ e)(a\ e)\text{ or }(a\ b)(b\ c)(c\ d)(d\ e).$$

In practice, in order to determine whether a given permutation is even or odd, one writes the permutation as a product of disjoint cycles. The permutation is odd if and only if this factorization contains an odd number of even-length cycles.

Another method for determining whether a given permutation is even or odd is to construct the corresponding permutation matrix and compute its determinant. The value of the determinant is the same as the parity of the permutation.

Every permutation of odd order must be even. The permutation (1 2)(3 4) in A4 shows that the converse is not true in general.

Equivalence of the two definitions
This section presents proofs that the parity of a permutation σ can be defined in two equivalent ways:

 * as the parity of the number of inversions in σ (under any ordering); or
 * as the parity of the number of transpositions that σ can be decomposed to (however we choose to decompose it).

Other definitions and proofs
The parity of a permutation of $$n$$ points is also encoded in its cycle structure.

Let σ = (i1 i2 ... ir+1)(j1 j2 ... js+1)...(ℓ1 ℓ2 ... ℓu+1) be the unique decomposition of σ into disjoint cycles, which can be composed in any order because they commute. A cycle (a b c ... x y z) involving k + 1 points can always be obtained by composing k transpositions (2-cycles):

 * $$(a\ b\ c \dots x\ y\ z)=(a\ b)(b\ c) \dots (x\ y)(y\ z),$$

so call k the size of the cycle, and observe that, under this definition, transpositions are cycles of size 1. From a decomposition into m disjoint cycles we can obtain a decomposition of σ into k1 + k2 + ... + km transpositions, where ki is the size of the ith cycle. The number N(σ) = k1 + k2 + ... + km is called the discriminant of σ, and can also be computed as

 * $$n \text{ minus the number of disjoint cycles in the decomposition of } \sigma$$

if we take care to include the fixed points of σ as 1-cycles.

Suppose a transposition (a b) is applied after a permutation σ. When a and b are in different cycles of σ then
 * $$(a\ b)(a\ c_1\ c_2 \dots c_r)(b\ d_1\ d_2 \dots d_s) = (a\ c_1\ c_2 \dots c_r\ b\ d_1\ d_2 \dots d_s)$$,

and if a and b are in the same cycle of σ then

 * $$(a\ b)(a c_1 c_2 \dots c_r\ b\ d_1\ d_2 \dots d_s) = (a\ c_1\ c_2 \dots c_r)(b\ d_1\ d_2 \dots d_s)$$.

In either case, it can be seen that N((a b)σ) = N(σ) ± 1, so the parity of N((a b)σ) will be different from the parity of N(σ).

If σ = t1t2 ... tr is an arbitrary decomposition of a permutation σ into transpositions, by applying the r transpositions $$t_1$$ after t2 after ... after tr after the identity (whose N is zero) observe that N(σ) and r have the same parity. By defining the parity of σ as the parity of N(σ), a permutation that has an even length decomposition is an even permutation and a permutation that has one odd length decomposition is an odd permutation.

 * Remarks:
 * A careful examination of the above argument shows that r ≥ N(σ), and since any decomposition of σ into cycles whose sizes sum to r can be expressed as a composition of r transpositions, the number N(σ) is the minimum possible sum of the sizes of the cycles in a decomposition of σ, including the cases in which all cycles are transpositions.
 * This proof does not introduce a (possibly arbitrary) order into the set of points on which σ acts.

Generalizations
Parity can be generalized to Coxeter groups: one defines a length function ℓ(v), which depends on a choice of generators (for the symmetric group, adjacent transpositions), and then the function v ↦ (−1)ℓ(v) gives a generalized sign map.

title: Parity of a permutation