Group commutator

The commutator $[𝑎, 𝑏]$ of two group members $𝑎, 𝑏 \in 𝐺$ is a way of describing how far two elements are from commuting with each other. For any $𝑎, 𝑏 \in 𝐺$ the commutator is defined as #m/def/group

[𝑎, 𝑏] = 𝑎 𝑏 𝑎 - 1 𝑏 - 1

Thus $𝑎$ and $𝑏$ commute iff $[𝑎, 𝑏] = 𝑒$ . The commutators of all elements forms a normal subgroup called the Commutator subgroup.

Properties

$[𝑏, 𝑎] = [𝑎, 𝑏] - 1$
$[𝑎 𝑏, 𝑐] = (𝑎 [𝑏, 𝑐]) [𝑎, 𝑐]$

Proof of 1–2

^P1 is obvious. Note the identity

[𝑎 𝑏, 𝑐] = 𝑎 𝑏 𝑐 𝑏 - 1 𝑎 - 1 𝑐 - 1 = 𝑎 𝑏 𝑐 𝑏 - 1 𝑐 - 1 𝑎 - 1 𝑎 𝑐 𝑎 - 1 𝑐 - 1 = (𝑎 [𝑏, 𝑐]) [𝑎, 𝑐]

proving ^P2.

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