# Exterior square preserves powering for nilpotent groups

From Groupprops

## Definition

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

## Facts used

- Schur multiplier preserves powering for nilpotent groups
- Derived subgroup is divisibility-closed in nilpotent group
- Powering is central extension-closed

## Proof

### Proof of powering

Recall that we have a short exact sequence, where denotes the Schur multiplier of :

where the image of in is a central subgroup of .

Fact (1) says that is -powered. Fact (2) says that is -powered. Fact (3) then gives us that is -powered.

### Proof of nilpotency

Since nilpotency is subgroup-closed, is nilpotent. Since is a central extension, it is also nilpotent. In fact, the nilpotency class of is at most . This follows from the fact that lower central series is strongly central applied to any Schur covering group of .