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