The Commutator of Two Elements in a Group

Definition: Let $G$ be a group. The Commutator of $g_1, g_2 \in G$ is defined to be $[g_1, g_2] := g_1g_2g_1^{-1}g_2^{-1}$.

Some people denote the commutator of $g_1, g_2 \in G$ instead by $[g_1, g_2] = g_1^{-1}g_2^{-1}g_1g_2$. Both definitions are equivalent.

The following proposition gives us some basic properties of commutators.

Proposition 1: Let $G$ be a group and let $g_1, g_2 \in G$. Then: a) $[g_1, g_2]^{-1} = [g_2, g_1]$. b) $g_2g_1g_2^{-1} = g_1^{-1}[g_1, g_2]$.

• Proof of a) If $g_1, g_2 \in G$ then:

(1)

\begin{align} \quad [g_1, g_2]^{-1} = (g_1g_2g_1^{-1}g_2^{-1})^{-1} = g_2g_1g_2^{-1}g_1^{-1} = [g_2, g_1] \quad \blacksquare \end{align}

• Proof of b) If $g_1, g_2 \in G$ then:

(2)

\begin{align} \quad g_2g_1g_2^{-1} = (g_1^{-1}g_1)g_2g_1g_2^{-1} = g_1^{-1}[g_1, g_2] \quad \blacksquare \end{align}