Finite groups can be generated by a pi-subgroup and a pi'-subgroup
arXiv:2103.17216
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.
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