A characterization of subgroup containment in direct products

Embedding Properties in Central Products

“…Hence, investigating subgroups that satisfy specific properties, known as characterizations, in products of groups is of particular interest. Edouard Goursat first determined how to identify and describe a subgroup of a direct product of two groups in [7], and his result has been refined as can be seen in [4]. Goursat's work provides the backbone necessary to understand subgroups of a direct product, and many mathematicians have since made significant contributions to this area.This article presents characterizations of subgroups in a central product of two groups, where a central product is a direct product with an amalgamated center.…”

“…Recently, more work has been done on subgroups of a direct product of two groups. For example, in [3], [4], [1], and [2], necessary and sufficient conditions have been provided to characterize subgroups of a direct product including normal and subnormal subgroups. However, very little work has been done to characterize these subgroups for a central product of two groups.…”