Elementarily equivalently embedded subgroups

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

Definition

Let G be a group and H_1 and H_2 be two subgroups. Consider the theory of the group G along with a rule for membership in H_1, and correspondingly consider a theory for G with a rule for membership in H_2. If these two theories are elementarily equivalent, then we say that H_1 and H_2 are elementarily equivalently embedded.

Relation with other relations

Stronger relations