# Exterior square preserves divisibility for nilpotent groups

## Statement

Suppose $G$ is a nilpotent group and $\pi$ is a set of prime numbers such that $G$ is $\pi$-divisible (i.e., $G$ is a $\pi$-divisible nilpotent group). The exterior square of $G$, denoted $G \wedge G$, is also a $\pi$-divisible nilpotent group.

## Proof

### Proof using the Schur covering group

Recall that the exterior square $G \wedge G$ can be defined as the derived subgroup of any Schur covering group of $G$. The Schur covering group is itself nilpotent (of class at most one more than the class of $G$) and is a stem extension of $G$. Facts (1) and (2) give the result.