# Elementarily equivalently embedded subgroups

From Groupprops

*This article defines an equivalence relation over the collection of subgroups within the same big group*

## Definition

Let be a group and and be two subgroups. Consider the theory of the group along with a rule for membership in , and correspondingly consider a theory for with a rule for membership in . If these two theories are elementarily equivalent, then we say that and are **elementarily equivalently embedded**.