Commutator subgroup

The commutator subgroup $[𝐺, 𝐺]$ is a normal subgroup of $𝐺$ generated by the group commutator of all elements, pairwise. #m/def/group

[𝐺, 𝐺] = ⟨ [𝑎, 𝑏] = 𝑎 𝑏 𝑎 - 1 𝑏 - 1 : 𝑎, 𝑏 \in 𝐺 ⟩ ⊴ 𝐺

Proof of normal subgroup

$[𝐺, 𝐺]$ is a subgroup by construction. Let $𝑔 \in [𝐺, 𝐺]$ . Then for any conjugate $𝑦 = 𝑥 𝑔 𝑥 - 1$ it follows $𝑦 𝑔 - 1 = 𝑥 𝑔 𝑥 - 1 𝑔 - 1 = [𝑥, 𝑔]$ , so $𝑦 𝑔 - 1 \in [𝐺, 𝐺]$ and thus $𝑦 𝑔 - 1 𝑔 = 𝑦 \in [𝐺, 𝐺]$ . Therefore $[𝐺, 𝐺]$ is a normal subgroup.

Wikipedia notes

[the commutator subgroup] is stable under every endomorphism of $𝐺$ : that is, $[𝐺, 𝐺]$ is a fully characteristic subgroup of $𝐺$ , a property considerably stronger than normality.

Properties

A quotient with the commutator subgroup of $𝐺$ is called an Abelianization of $𝐺$ .

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