Monoid of Self-Inverse Elements is Abelian Group

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Theorem

Let $\struct {S, \circ}$ be a monoid such that:

$\forall x \in S: x \circ x = e$

where $e$ is the identity element of $\struct {S, \circ}$.


Then $\struct {S, \circ}$ is an abelian group.


Proof

From Equivalence of Definitions of Self-Inverse, $x \circ x = e \implies x = x^{-1}$.

From Inverse in Monoid is Unique, it follows that every element of $\struct {S, \circ}$ has a unique inverse.

So by definition, $\struct {S, \circ}$ is a group.


From All Elements Self-Inverse then Abelian, it follows that $\struct {S, \circ}$ is an abelian group.

$\blacksquare$


Sources