Group theory MOC

Group commutator

The commutator [𝑎,𝑏] of two group members 𝑎,𝑏 ∈𝐺 is a way of describing how far two elements are from commuting with each other. For any 𝑎,𝑏 ∈𝐺 the commutator is defined as group

[𝑎,𝑏]=𝑎𝑏𝑎−1𝑏−1

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

Properties

1. [𝑏,𝑎] =[𝑎,𝑏]−1
2. [𝑎𝑏,𝑐] =(𝑎[𝑏,𝑐])[𝑎,𝑐]

Proof of 1–2

^P1 is obvious. Note the identity

[𝑎𝑏,𝑐]=𝑎𝑏𝑐𝑏−1𝑎−1𝑐−1=𝑎𝑏𝑐𝑏−1𝑐−1𝑎−1𝑎𝑐𝑎−1𝑐−1=(𝑎[𝑏,𝑐])[𝑎,𝑐]

proving ^P2.

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