Elementarily equivalently embedded subgroups

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

• Conjugate subgroups
• Automorphic subgroups