# Centers of Sylow Subgroups and Automorphisms

## TL;DR

This paper proves that for odd primes p, automorphisms of p-power order centralizing Sylow p-subgroups are inner in certain finite groups, confirming a conjecture and exploring implications for the structure of these groups.

## Contribution

It establishes that automorphisms of p-power order centralizing Sylow p-subgroups are inner in groups with no non-trivial normal p'-subgroups, confirming Gross's conjecture.

## Key findings

- Automorphisms of p-power order centralizing Sylow p-subgroups are inner for odd primes p.
- The center of Sylow p-subgroups is contained in the generalized Fitting subgroup.
- For p=2, the square of such automorphisms is inner, as proved by Glauberman.

## Abstract

Suppose that p is an odd prime and G is a finite group having no normal non-trivial p'-subgroup. We show that if a is an automorphism of G of p-power order centralizing a Sylow p-group of G, then a is inner. This answers a conjecture of Gross. An easy corollary is that if p is an odd prime and P is a Sylow p-subgroup of G, then the center of P is contained in the generalized Fitting subgroup of G. We give two proofs both requiring the classification of finite simple groups. For p=2, the result fails but Glauberman in 1968 proved that the square of a is inner. This answered a problem of Kourovka posed in 1999.