Superperfectness is finite direct product-closed

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This article gives the statement, and possibly proof, of a group property (i.e., superperfect group) satisfying a group metaproperty (i.e., finite direct product-closed group property)
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Statement

Statement for two groups

Suppose G_1 and G_2 are superperfect groups (possibly isomorphic, possibly non-isomorphic). Then, the external direct product G_1 \times G_2 is also a superperfect group.

Statement for finitely many groups

Suppose n > 1 with n a positive integer. Suppose G_1,G_2,\dots,G_n are superperfect groups (possibly isomorphic, possibly non-isomorphic). Then, the external direct product G_1 \times G_2 \times \dots \times G_n is also a superperfect group.

Examples

Since the smallest nontrivial superperfect group is SL(2,5) which has order 120, the smallest nontrivial example of a superperfect group obtained as a direct product of smaller superperfect groups is the direct product of SL(2,5) and SL(2,5), which is a group of order 14400.

Facts used

  1. Kunneth formula for group homology

Proof

Proof for two groups

The proof basically follows from Fact (1), which allows us to compute the homology groups of a direct product in terms of the homology groups of the direct factors.

Proof for finitely many groups

This follows from the proof for two groups and using mathematical induction.