Finite groups can be generated by a pi-subgroup and a pi'-subgroup
Group Theory
2023-06-19 v4
Abstract
Answering a question of Dan Haran and generalizing some results of Aschbacher-Guralnick and Suzuki, we prove that given a set of primes pi, any finite group can be generated by a pi-subgroup and a pi'-subgroup. This gives a free product description of a countably generated free profinite group.
Keywords
Cite
@article{arxiv.2103.17216,
title = {Finite groups can be generated by a pi-subgroup and a pi'-subgroup},
author = {Thomas Breuer and Robert M. Guralnick},
journal= {arXiv preprint arXiv:2103.17216},
year = {2023}
}
Comments
We generalized the main result in the previous version, simplified the proof and give an application to profinite groups. In the latest version, we also proved a result about generation by Sylow subgroups. The paper will appear in the Israel Journal of Mathematics