Group Element Commutes with Inverse

Theorem

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $x \in G$.

Then:

$x \circ x^{-1} = x^{-1} \circ x$

That is, $x$ commutes with its inverse $x^{-1}$.

Proof

By definition of inverse element:

$x \circ x^{-1} = e = x^{-1} \circ x$

Hence the result by definition.

Sources

• 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 2$: The Axioms of Group Theory
• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.6$. Elementary theorems on groups: Example $86$