Conjugate subgroup

Two subgroups $𝐻, 𝐾 \subseteq 𝐺$ are conjugate to each other iff there exists $𝑔 \in 𝐺$ such that #m/def/group

𝐾 = 𝑔 𝐻 𝑔 - 1 = {𝑔 ℎ 𝑔 - 1 : ℎ \in 𝐻}

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 automorphism on the supergroup.

#state/tidy | #lang/en | #SemBr