Finite groups can be generated by a pi-subgroup and a pi'-subgroup

TL;DR

This paper proves that any finite group can be generated by a pi-subgroup and a pi'-subgroup, generalizing previous results and providing a new perspective on group generation and free profinite groups.

Contribution

It introduces a new generation method for finite groups using pi- and pi'-subgroups, extending prior theoretical frameworks.

Findings

Any finite group can be generated by a pi- and a pi'-subgroup.

Provides a free product description of a countably generated free profinite group.

Generalizes results of Aschbacher-Guralnick and Suzuki.

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.