单词、置换和有限群的不可解长度

Pub Date : 2019-04-04 DOI:10.4171/JCA/51
Alexander Bors, A. Shalev
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引用次数: 3

摘要

研究了某些恒等式和概率恒等式对有限群结构的影响。更具体地说,设$w$为$d$不同变量中的一个非平凡单词,设$G$为一个有限组,其中对于某个固定的$\rho>0$,单词映射$w_G:G^d\rightarrow G$的纤维大小至少为$\rho|G|^d$。我们表明,对于某些单词$w$,这意味着$G$有一个正常可解的子群,该子群在$w$和$\rho$方面有界。我们还表明,对于更大的单词族$w$,这意味着$G$的不可解长度在$w$和$\rho$方面有界,从而为Larsen的猜想提供了证据。在此过程中,我们获得了一些独立的结果,大致表明大型有限置换群的大多数元素具有很大的支持度。
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Words, permutations, and the nonsolvable length of a finite group
We study the impact of certain identities and probabilistic identities on the structure of finite groups. More specifically, let $w$ be a nontrivial word in $d$ distinct variables and let $G$ be a finite group for which the word map $w_G:G^d\rightarrow G$ has a fiber of size at least $\rho|G|^d$ for some fixed $\rho>0$. We show that, for certain words $w$, this implies that $G$ has a normal solvable subgroup of index bounded above in terms of $w$ and $\rho$. We also show that, for a larger family of words $w$, this implies that the nonsolvable length of $G$ is bounded above in terms of $w$ and $\rho$, thus providing evidence in favor of a conjecture of Larsen. Along the way we obtain results of some independent interest, showing roughly that most elements of large finite permutation groups have large support.
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