### 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).

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).

#### Full Text:

PDFDOI: https://doi.org/10.2478/tatra.v64i0.374