# Definition:Self-Inverse Element/Definition 1

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## Contents

## Definition

Let $\struct {S, \circ}$ be a monoid whose identity element is $e$.

Let $x \in S$ be an element of $S$.

$x$ is a **self-inverse element of $\struct {S, \circ}$** if and only if $x \circ x = e$.

## Also known as

The definition of a **self-inverse element** is usually made in the context of a group.

Some sources refer to such an element as an **involution**.

## Also see

## Sources

- 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $1$: Introduction to Finite Group Theory: $1.13$