The soluble radical and orbits of certain maps on finite groups

2020/10/08 by Popović, David, Wilson, John S.
#20D05 #20D10 #20F45 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2010.03944

Abstract

For each element u in a finite group G define a map θ_u\colon G→ G by θ_u(g)=[g^-u,g] and set Θ_G(u)=\g∈ G| θ_uⁿ(g)=g \hbox for some n>0\. Then θ_u induces a permutation of Θ_G(u); let β_G(u) be the number of orbits apart from \1\. Building on work of J.N. Bray, R.A. Wilson and the second author, we show that the index of the soluble radical of a finite group G is bounded in terms of the values of β_G(u) for 2-elements u.

Related

• Automorphism orbits and element orders in finite groups: almost-solubility and the Monster
• Cycle lengths in finite groups and the size of the solvable radical
• Criteria for solvable radical membership via p-elements
• On the soluble graph of a finite group
• Finite groups admitting a coprime automorphism satisfying an additional polynomial identity