Exterior square of finite group is finite

Statement

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

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

Proof

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