Finite groups with a large automorphism orbit

2018/02/26 by Bors, Alexander
#20D60 #60C05 #FOS: Mathematics #Group Theory (math.GR) #Primary: 20D45. Secondary: 20D05

paper · doi:10.48550/arxiv.1802.09215

Abstract

We study the nonabelian composition factors of a finite group G assumed to admit an Aut(G)-orbit of length at least ρ|G|, for a given ρ∈(0,1]. Our main results are the following: The orders of the nonabelian composition factors of G are then bounded in terms of ρ, and if ρ>(18)/(19), then G is solvable. On the other hand, for each nonabelian finite simple group S, there is a constant c(S)∈(0,1] such that S occurs with arbitrarily large multiplicity as a composition factor in some finite group G having an Aut(G)-orbit of length at least c(S)|G|.

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