# Coleman automorphisms of finite groups and their minimal normal subgroups

## TL;DR

This paper investigates Coleman automorphisms in finite groups, proving they are inner under certain conditions, and explores their properties in various group constructions, providing partial answers to existing open questions.

## Contribution

It establishes conditions under which Coleman automorphisms are inner and characterizes them in specific group extensions, advancing understanding in automorphism theory.

## Key findings

- All Coleman automorphisms are inner for groups with self-central minimal characteristic subgroups.
- Holomorphs and wreath products of finite simple groups have no non-inner Coleman automorphisms.
- Class-preserving automorphisms of certain nilpotent-by-nilpotent groups are inner.

## Abstract

In this paper, we show that all Coleman automorphisms of a finite group with self-central minimal non-trivial characteristic subgroup are inner; therefore the normalizer property holds for these groups. Using our methods we show that the holomorph and wreath product of finite simple groups, among others, have no non-inner Coleman automorphisms. As a further application of our theorems, we provide partial answers to questions raised by M. Hertweck and W. Kimmerle. Furthermore, we characterize the Coleman automorphisms of extensions of a finite nilpotent group by a cyclic $p$-group. Lastly, we note that class-preserving automorphisms of 2-power order of some nilpotent-by-nilpotent groups are inner, extending a result by J. Hai and J. Ge.