Exterior square of finite group is finite

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Statement

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

Applications

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).