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

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