SOME OPEN PROBLEMS IN THE THEORY OF INFINITE DIMENSIONAL ALGEBRAS
SOME OPEN PROBLEMS IN THE THEORY OF INFINITE DIMENSIONAL ALGEBRAS
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无限维代数理论中的一些悬而未决的问题
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发表时间：
2007
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通讯作者：
E. Zelmanov
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作者：
E. Zelmanov
我们将讨论有关无限维代数的一些很老的和一些新的公开问题。所有这些问题都受到组合群论的启发。I. 1902年W.伯恩赛德提出了他著名的挠群问题：（1）设G是n-生成挠群，即对任意的元素g ∈ G存在n = n（g）> 1，使得g = 1。这是否意味着G是有限的？(2)设群G是n-生成的，且有界度的挠，即存在n > 1使得对任意元素g ∈ G，g = 1.这是否意味着G是有限的？W.伯恩赛德[7]和我。Schur [43]对线性群证明了（1）。对于n = 2，3（W.伯恩赛德，[6]），n = 4（I. N. Sanov，[42]）和n = 6（M. Hall，[17]）。1964年E. S.戈洛和我。R. Shafarevich（[12]，[13]）对任意素数p构造了一族无限n-生成p-群（对任意元素g存在n = n（g）> 1使得gpn = 1），这是对问题（1）的否定回答。其他的非生成扭群是由S. V. Alyoshin [1]，R. I. Grigorchuk，N.古普塔-S。Sidki，V. I. Sushchansky [48]. 1968年，P.S. Novikov和S. I. Adian构造了有界奇度n > 4381的无限n-生成群。1994年S. Ivanov [19]将其推广到n = 2，k > 32，所以现在我们可以说问题（2）对所有充分大的n都有负解。但请注意，上面所有的反例都没有列出。以下重要问题仍然悬而未决。问题1.是否存在无限的双表示挠群？2006年10月28日收到。2000年数学学科分类。十七二十
We will discuss some very old and some new open problems concerning infinite dimensional algebras. All these problems have been inspired by combinatorial group theory. I. The Burnside and Kurosh problems In 1902 W. Burnside formulated his famous problems for torsion groups: (1) let G be a finitely generated torsion group, that is, for an arbitrary element g ∈ G there exists n = n(g) > 1, such that g = 1. Does it imply that G is finite? (2) Let a group G be finitely generated and torsion of bounded degree, that is, there exists n > 1 such that for an arbitrary element g ∈ G g = 1. Does it imply that G is finite? W. Burnside [7] and I. Schur [43] proved (1) for linear groups. The positive answer for (2) is known for n = 2, 3 (W. Burnside, [6]), n = 4 (I. N. Sanov, [42]) and n = 6 (M. Hall, [17]). In 1964 E. S. Golod and I. R. Shafarevich ([12], [13]) constructed a family of infinite finitely generated p–groups (for an arbitrary element g there exists n = n(g) > 1 such that gpn = 1) for an arbitrary prime p. This was a negative answer to the question (1). Other finitely generated torsion groups were constructed by S. V. Alyoshin [1], R. I. Grigorchuk [14], N. Gupta –S. Sidki [16], V. I. Sushchansky [48]. In 1968 P. S. Novikov and S. I. Adian constructed infinite finitely generated groups of bounded odd degree n > 4381. In 1994 S. Ivanov [19] extended this to n = 2, k > 32, so now we can say that the question (2) has negative solution for all sufficiently large n. Remark though that all the counterexamples above are not finitely presented. The following important problem still remains open. Problem 1. Do there exist infinite finitely presented torsion groups? Received October 28, 2006. 2000 Mathematics Subject Classification. 17, 20.