Large characteristic subgroups satisfying multilinear commutator identities

2007/06/01 by E. I. Khukhro, N. Yu. Makarenko · 2 citations
Mathematics · Engineering · Computer Science · #Finite Group Theory Research #graph theory and CDMA systems #Coding theory and cryptography

paper · doi:10.1112/jlms/jdm047

Abstract

It is proved that if a group G has a subgroup H of finite index n satisfying a multilinear commutator identity ϰ (H) = 1, then G has also a characteristic subgroup C satisfying the same identity ϰ (C) = 1 and having finite index bounded in terms of n and the weight of ϰ. As an illustration, the result on characteristic subgroups is applied to a finite group with an almost regular automorphism of order 4 to obtain a subgroup S of bounded index with a ‘strong’ bound for the nilpotency class of γ3(S).

Cited by

• Khukhro-Makarenko type theorems for algebras
• Invariant systems of weighted representatives

Related

• On profinite groups with commutators covered by nilpotent subgroups
• Finite groups whose maximal subgroups have almost odd index
• BFC-theorems for higher commutator subgroups
• On finite groups whose Sylow subgroups have a bounded number of generators
• Groups with finitely many long commutators of maximal order