翻訳と辞書 Words near each other ・ Burnside theorem ・ Burnside Township ・ Burnside Township, Centre County, Pennsylvania ・ Burnside Township, Clearfield County, Pennsylvania ・ Burnside Township, Goodhue County, Minnesota ・ Burnside Township, Michigan ・ Burnside Township, Pennsylvania ・ Burnside Township, Webster County, Iowa ・ Burnside Triangle ・ Burnside Village ・ Burnside War Memorial Hospital ・ Burnside West Christchurch University Cricket Club ・ Burnside's Bridge ・ Burnside's lemma ・ Burnside's North Carolina Expedition ・ Burnside's problem ・ Burnside, Arizona ・ Burnside, Canterbury ・ Burnside, Chicago ・ Burnside, Colchester County ・ Burnside, County Antrim ・ Burnside, Highland ・ Burnside, Illinois ・ Burnside, Iowa ・ Burnside, Kentucky ・ Burnside, Louisiana ・ Burnside, Mississippi ・ Burnside, Newfoundland and Labrador ・ Burnside, Nova Scotia ・ Burnside, Otago Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Burnside's problem ： ウィキペディア英語版 Burnside's problem The Burnside problem, posed by William Burnside in 1902 and one of the oldest and most influential questions in group theory, asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. In plain language, if by looking at individual elements of a group we suspect that the whole group is finite, must it indeed be true? The problem has many variants (see bounded and restricted below) that differ in the additional conditions imposed on the orders of the group elements. == Brief history == Initial work pointed towards the affirmative answer. For example, if a group ''G'' is generated by ''m'' elements and the order of each element of ''G'' is a divisor of 4, then ''G'' is finite. Moreover, A. I. Kostrikin was able to prove in 1958 that among the finite groups with a given number of generators and a given prime exponent, there exists a largest one. This provides a solution for the restricted Burnside problem for the case of prime exponent. (Later in 1989 Efim Zelmanov was able to solve the restricted Burnside problem for an arbitrary exponent.) Issai Schur had showed in 1911 that any finitely generated periodic group that was a subgroup of the group of invertible ''n'' × ''n'' complex matrices was finite; he used this theorem to prove the Jordan–Schur theorem. Nevertheless, the general answer to Burnside's problem turned out to be negative. In 1964, Golod and Shafarevich constructed an infinite group of Burnside type without assuming that all elements have uniformly bounded order. In 1968, Pyotr Novikov and Sergei Adian's supplied a negative solution to the bounded exponent problem for all odd exponents larger than 4381. In 1982, A. Yu. Ol'shanskii found some striking counterexamples for sufficiently large odd exponents (greater than 1010), and supplied a considerably simpler proof based on geometric ideas. The case of even exponents turned out to be much harder to settle. In 1992 S. V. Ivanov announced the negative solution for sufficiently large even exponents divisible by a large power of 2 (detailed proofs were published in 1994 and occupied some 300 pages). Later joint work of Ol'shanskii and Ivanov established a negative solution to an analogue of Burnside's problem for hyperbolic groups, provided the exponent is sufficiently large. By contrast, when the exponent is small and different from 2,3,4 and 6, very little is known. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Burnside's problem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.