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arXiv · 2111.12534

Flexibility in generating sets of finite groups

Scott Harper
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Abstract

Let G be a finite group. It has recently been proved that every nontrivial element of G is contained in a generating set of minimal size if and only if all proper quotients of G require fewer generators than G. It is natural to ask which finite groups, in addition, have the property that any two elements of G that do not generate a cyclic group can be extended to a generating set of minimal size. This note answers the question. The only such finite groups are very specific affine groups: elementary abelian groups extended by a cyclic group acting as scalars.

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Scott Harper. 2021-11-24. Flexibility in generating sets of finite groups. https://arxiv.org/abs/2111.12534

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