Exterior square preserves divisibility for nilpotent groups

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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.

Facts used

  1. Stem extension preserves divisibility for nilpotent groups
  2. Derived subgroup is divisibility-closed in 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.