Exterior square preserves powering for nilpotent groups

Definition

Suppose $\pi$ is a set of primes and $G$ is a $\pi$-powered nilpotent group, i.e., $G$ is both a nilpotent group and a $\pi$-powered group. Then, the exterior square $G \wedge G$ is also a $\pi$-powered nilpotent group.

Proof

Proof of powering

Recall that we have a short exact sequence, where $M(G)$ denotes the Schur multiplier of $G$: $0 \to M(G) \to G \wedge G \to [G,G] \to 1$

where the image of $M(G)$ in $G \wedge G$ is a central subgroup of $G \wedge G$.

Fact (1) says that $M(G)$ is $\pi$-powered. Fact (2) says that $[G,G]$ is $\pi$-powered. Fact (3) then gives us that $G \wedge G$ is $\pi$-powered.

Proof of nilpotency

Since nilpotency is subgroup-closed, $[G,G]$ is nilpotent. Since $G \wedge G$ is a central extension, it is also nilpotent. In fact, the nilpotency class of $G \wedge G$ is at most $\lceil c/2 \rceil$. This follows from the fact that lower central series is strongly central applied to any Schur covering group of $G$.