# Superperfectness is finite direct product-closed

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.