Supersoluble groups and the probability of generating a supersoluble subgroup

Andrea Lucchini

Supersoluble groups and the probability of generating a supersoluble subgroup

Group Theory 2026-07-09 v1

Authors: Andrea Lucchini

Abstract

Let $G$ be a finite group and let $\mathrm{P}_{\mathcal{U}}(G)$ denote the probability that two randomly chosen elements of $G$ generate a supersoluble subgroup. We prove that if $\mathrm{P}_{\mathcal{U}}(G) \geq 16/25$ then $G$ is supersoluble, and that the bound $16/25$ is sharp, being attained by the group $G = (C_5 \times C_5) \rtimes Q_8$ , where $Q_8$ acts faithfully and irreducibly on $C_5 \times C_5$ .

Keywords

supersoluble group finite group generating probability group theory sharp bound

Cite

@article{arxiv.2607.08165,
  title  = {Supersoluble groups and the probability of generating a supersoluble subgroup},
  author = {Andrea Lucchini},
  journal= {arXiv preprint arXiv:2607.08165},
  year   = {2026}
}