# IA equals inner in extraspecial

## Contents

## Statement

In an Extraspecial group (?) (a group of prime power order whose commutator subgroup, center and Frattini subgroup are all equal and cyclic) any IA-automorphism (?) is inner.

## Definitions used

### Extraspecial group

`Further information: extraspecial group`

A group of prime power order is termed an **extraspecial group** if its commutator subgroup, center, and Frattini subgroup are all equal, and this subgroup is a cyclic group of prime order.

### IA-automorphism

`Further information: IA-automorphism`

An IA-automorphism of a group is an automorphism that induces the identity automorphism on the Abelianization of the group (the quotient by its commutator subgroup).

## Related facts

### An equivalent fact

## Proof

*Given*: An extraspecial -group . Let denote the group of IA-automorphisms of , and denote the group of inner automorphisms of

*To prove*:

*Proof*: Note that (i.e., any inner automorphisms is IA). Also, (again, a standard fact). Since both groups are finite, it suffices to show that .

The first step in this is to show that elements of act as the identity. not only on the quotient , but also on . Thus, can be viewed as the stability group of the series . So, what we need to prove is that the cardinality of this stability group is at most .