Subgroup

Conjugate subgroup

Two subgroups 𝐻,𝐾 ⊆𝐺 are conjugate to each other iff there exists 𝑔 ∈𝐺 such that group

𝐾=𝑔𝐻𝑔−1={𝑔ℎ𝑔−1:ℎ∈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.

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