Finite groups with an automorphism inverting, squaring or cubing a\n non-negligible fraction of elements

2016/01/17 by Alexander Bors, Bors, Alexander
Mathematics · #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.1601.04311

Abstract

There are various results in the literature which are part of the general\nphilosophy that a finite group for which a certain parameter (for example, the\nnumber of conjugacy classes or the maximum number of elements inverted, squared\nor cubed by a single automorphism) is large enough must be close to being\nabelian. In this paper, we show the following: Fix a real number \ρ with\n0<\ρ\≤ 1. Then a finite group G with an automorphism inverting or\nsquaring at least \ρ|G| of the elements in G is "almost abelian" in the\nsense that both the index and the derived length of the solvable radical of G\nare bounded. Furthermore, if G has an automorphism cubing at least \ρ|G|\nof the elements in G, then G is "almost solvable" in the sense that the\nindex of the solvable radical of G is bounded.\n

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