# Exterior square of finite group is finite

From Groupprops

## Contents

## Statement

Suppose is a finite group. Then, the exterior square of is also a finite group.

## Applications

- Schur-Baer theorem: This states that if the inner automorphism group of a group is finite, so is the derived subgroup of the group.

## Facts used

- Commutator map is homomorphism from exterior square to derived subgroup, and the kernel of this homomorphism is the Schur multiplier.
- Schur multiplier of finite group is finite

## Proof

The proof basically follows directly by combining Facts (1) and (2).