A generalized Goursat lemma

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 \ $A_1\times A_2 \times \cdots \times A_n$ \ 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).