# Exterior square preserves divisibility for nilpotent groups

From Groupprops

## Statement

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

## Facts used

- Stem extension preserves divisibility for nilpotent groups
- Derived subgroup is divisibility-closed in nilpotent group

## Proof

### Proof using the Schur covering group

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