Characteristic classes of involutions in nonsolvable groups

Abstract

Abstract Let G , D 0 , D 1 {G,D_{0},D_{1}} be finite groups such that D 0 ⁢ ⊴ ⁢ D 1 {D_{0}\trianglelefteq D_{1}} are groups of automorphisms of G that contain the inner automorphisms of G. Assume that D 1 / D 0 {D_{1}/D_{0}} has a normal 2-complement and that D 1 {D_{1}} acts fixed-point-freely on the set of D 0 {D_{0}} -conjugacy classes of involutions of G (i.e., C D 1 ⁢ ( a ) ⁢ D 0 < D 1 {C_{D_{1}}(a)D_{0}<D_{1}} for every involution a ∈ G {a\in G} ). We prove that G is solvable. We also construct a nonsolvable finite group that possesses no characteristic conjugacy class of nontrivial cyclic subgroups. This shows that an assumption on the structure of D 1 / D 0 {D_{1}/D_{0}} above must be made in order to guarantee the solvability of G and also yields a negative answer to Problem 3.51 in the Kourovka notebook, posed by A. I. Saksonov in 1969.