Exterior square preserves powering for nilpotent groups

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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.

Facts used

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

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.