[[Subgroup]]
# Conjugate subgroup

Two subgroups $H, K \sube G$ are **conjugate** to each other iff there exists $g \in G$ such that #m/def/group
$$
\begin{align*}
K = gHg^{-1} = \{ ghg^{-1} : h \in H \}
\end{align*}
$$
This is a natural application of the [[Conjugation by an element]] to entire subgroup.
If a subgroup is only conjugate to itself, it is called a [[Normal subgroup]].

## Properties

- Conjugate subgroups are isomorphic to each other, since conjugation is itself an [[Morphism|automorphism]] on the supergroup.

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