A Generalized Goursat Lemma

TL;DR

This paper extends the Goursat lemma to describe subgroups of direct products of multiple groups, with applications to various group types including profinite, cyclic, and Sylow p-subgroups.

Contribution

It provides a generalized form of the Goursat lemma for multiple groups and explores applications to complex group structures.

Findings

Generalization of Goursat lemma to n-fold direct products

Applications to profinite, cyclic, and Sylow p-groups

Deeper insights into subgroup characterizations in complex groups

Abstract

In this note the usual Goursat lemma, which describes subgroups of the direct product of two groups, is generalized to describing subgroups of a direct product $A_1\times A_2 \times...\times A_n$ of a finite number of groups. Other possible generalizations are discussed and applications characterizing several types of subgroups are given. Most of these applications are straightforward, while somewhat deeper applications occur in the case of profinite groups, cyclic groups, and the Sylow $p$-subgroups (including infinite groups that are virtual $p$-groups).