A note on finite groups with an automorphism inverting or squaring a\n non-negligible fraction of elements

2016/06/02 by Alexander Bors, Bors, Alexander
Mathematics · Computer Science · #Finite Group Theory Research #Limits and Structures in Graph Theory #Coding theory and cryptography

paper · pdf · doi:10.48550/arxiv.1606.00635

Abstract

We show that for a finite group G, the commuting probability of G can be\nexplicitly bounded from below in a nontrivial way by a function in the maximum\nfraction of elements inverted resp. squared by an automorphism of G. Using\nthese bounds together with a result of Guralnick and Robinson gives upper\nbounds on the index of the Fitting subgroup of G under each of the two\nconditions that G have an automorphism inverting resp. squaring at least\n\ρ|G| many elements in G, for \ρ\∈\(0,1\] fixed. This is an\nimprovement on previous results of the author.\n

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