Subgroups of 3-factor direct products

TL;DR

This paper extends Goursat's Lemma to analyze the structure of subdirect products in three-factor direct products, revealing that dependencies among elements are Abelian and providing enumeration formulas.

Contribution

It introduces a structure theorem for three-factor subdirect products and generalizes Goursat's Lemma to this setting, including explicit counting methods.

Findings

Dependencies among elements are Abelian in nature.

Provides a structure theorem for subdirect products.

Derives an explicit formula for counting subdirect products of symmetric groups.

Abstract

Extending Goursat's Lemma we investigate the structure of subdirect products of 3-factor direct products. We give several example constructions and then provide a structure theorem showing that every such group is essentially obtained by a combination of the constructions. The central observation in this structure theorem is that the dependencies among the group elements in the subdirect product that involve all three factors are of Abelian nature. In the spirit of Goursat's Lemma, for two special cases, we derive correspondence theorems between data obtained from the subgroup lattices of the three factors (as well as isomorphism between arising factor groups) and the subdirect products. Using our results we derive an explicit formula to count the number of subdirect products of the direct product of three symmetric groups.