Probabilistically nilpotent groups of class two

Abstract

For $G$ a finite group, let $d_2(G)$ denote the proportion of triples $(x,y,z)\in G^3$ such that $[x,y,z]=1$. We determine the structure of finite groups $G$ such that $d_2(G)$ is bounded away from zero: if $d_2(G)\ge \epsilon >0$, $G$ has a class-4 nilpotent normal subgroup $H$ such that $[G:H]$ and $|{\gamma }_4(H)|$ are both bounded in terms of $\epsilon$. We also show that if $G$ is an infinite group whose commutators have boundedly many conjugates, or indeed if $G$ satisfies a certain more general commutator covering condition, then $G$ is finite-by-class-3-nilpotent-by-finite.

20 pages

References in corpus (1)

• Biased multilinear maps of abelian groups

Cited by in corpus (1)

• On finite groups with bounded conjugacy classes of commutators