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Coprime invariable generation and minimal-exponent groups

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Author(s)

Eloisa Detomi, Andrea Lucchini, C.M. Roney-Dougal

School/Research organisations

Abstract

A finite group G is coprimely invariably generated if there exists a set of generators {g1,. .,gu} of G with the property that the orders |g1|,. .,|gu| are pairwise coprime and that for all x1,. .,xu∈G the set {g1x1,. .,guxu} generates G.

We show that if G is coprimely invariably generated, then G can be generated with three elements, or two if G is soluble, and that G has zero presentation rank. As a corollary, we show that if G is any finite group such that no proper subgroup has the same exponent as G, then G has zero presentation rank. Furthermore, we show that every finite simple group is coprimely invariably generated by two elements, except for O8+(2) which requires three elements.

Along the way, we show that for each finite simple group S, and for each partition π1,. .,πu of the primes dividing |S|, the product of the number kπi(S) of conjugacy classes of πi-elements satisfies. ∏i=1u kπi(S)≤|S|/2|OutS|.
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Details

Original languageEnglish
Pages (from-to)3453-3465
Number of pages13
JournalJournal of Pure and Applied Algebra
Volume219
Issue number8
Early online date12 Dec 2014
DOIs
Publication statusPublished - Aug 2015

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