Finite groups satisfying the independence property

Saul Daniel Freedman, Andrea Lucchini*, Daniele Nemmi, Colva Mary Roney-Dougal

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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Abstract

We say that a finite group G satisfies the independence property if, for every pair of distinct elements x and y of G, either {x, y} is contained in a minimal generating set for G or one of x and y is a power of the other. We give a complete classification of the finite groups with this property, and in particular prove that every such group is supersoluble. A key ingredient of our proof is a theorem showing that all but three finite almost simple groups H contain an element s such that the maximal subgroups of H containing s, but not containing the socle of H, are pairwise non-conjugate.
Original languageEnglish
Pages (from-to)509-545
Number of pages37
JournalInternational Journal of Algebra and Computation
Volume33
Issue number3
Early online date29 Apr 2023
DOIs
Publication statusPublished - 1 May 2023

Keywords

  • Generating sets
  • Supersoluble groups
  • Simple groups

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