Coprime invariable generation and minimal-exponent groups
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|.
Citation
Detomi , E , Lucchini , A & Roney-Dougal , C M 2015 , ' Coprime invariable generation and minimal-exponent groups ' , Journal of Pure and Applied Algebra , vol. 219 , no. 8 , pp. 3453-3465 . https://doi.org/10.1016/j.jpaa.2014.12.005
Publication
Journal of Pure and Applied Algebra
Status
Peer reviewed
ISSN
0022-4049Type
Journal article
Description
Colva Roney-Dougal acknowledges the support of EPSRC grant EP/I03582X/1.Collections
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