Engel sinks of fixed points in finite groups

2018/09/08 by Acciarri, Cristina, Shumyatsky, Pavel, da Silveira, Danilo Sanção
#20D10 #20D45 #20F45 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1809.02733

Abstract

For an element g of a group G, an Engel sink is a subset \mathscrE(g) such that for every x∈ G all sufficiently long commutators [x,g,g,…,g] belong to \mathscrE(g). Let q be a prime, let m be a positive integer and A an elementary abelian group of order q² acting coprimely on a finite group G. We show that if for each nontrivial element a in A and every element g∈ C_G(a) the cardinality of the smallest Engel sink \mathscrE(g) is at most m, then the order of γ_∞(G) is bounded in terms of m only. Moreover we prove that if for each a∈ A∖ \1\ and every element g∈ C_G(a), the smallest Engel sink \mathscrE(g) generates a subgroup of rank at most m, then the rank of γ_∞(G) is bounded in terms of m and q only.

Related

• On profinite groups with automorphisms whose fixed points have countable Engel sinks
• On finite groups with an automorphism of prime order whose fixed points have bounded Engel sinks
• Finite groups with Engel sinks of bounded rank
• On finite groups with automorphisms whose fixed points are Engel
• Compact groups with countable Engel sinks