Equivalence of definitions of conjugate subgroups

This article gives a proof/explanation of the equivalence of multiple definitions for the term conjugate subgroups
View a complete list of pages giving proofs of equivalence of definitions

Statement

Let $G$ be a group and $H,K$ be subgroups. Then, the following are equivalent:

1. $H$ and $K$ are conjugate subgroups in $G$: there exists $g\in G$ such that $K=gHg^{-1}$
2. There exists a transitive group action of $G$ on a nonempty set $S$, and (possibly identical) points $s,t\in S$ such that $H,K$ are the isotropy subgroups at $s$ and $t$ respectively.
3. There exists a group action of $G$ on a set $S$, and (possibly identical) points $s,t\in S$ that are in the same orbit under the $G$-action, such that $H,K$ are the isotropy subgroups at $H$ and $K$.