Words, permutations, and the nonsolvable length of a finite group

2019/04/04 by Bors, Alexander, Shalev, Aner
#20B05 #20D06 #20E10 #20F22 (Secondary) #20P05 (Primary) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1904.02370

Abstract

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→ G has a fiber of size at least ρ|G|^d for some fixed ρ>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 ρ. 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 ρ, 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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