# Group is Inverse Semigroup with Identity

## Theorem

A group is an inverse semigroup with an identity.

## Proof

Let $\left({S, \circ}\right)$ be a group. Then for all $a \in S$:

 $\displaystyle e$ $=$ $\displaystyle a \circ a^{-1}$ Every $a \in G$ is Invertible $\displaystyle \implies \ \$ $\displaystyle e \circ a$ $=$ $\displaystyle a \circ a^{-1} \circ a$ $\displaystyle \implies \ \$ $\displaystyle a$ $=$ $\displaystyle a \circ a^{-1} \circ a$ Definition of Identity

and

 $\displaystyle e$ $=$ $\displaystyle a \circ a^{-1}$ Every $a \in G$ is Invertible $\displaystyle \implies \ \$ $\displaystyle a^{-1} \circ e$ $=$ $\displaystyle a^{-1} \circ a \circ a^{-1}$ $\displaystyle \implies \ \$ $\displaystyle a^{-1}$ $=$ $\displaystyle a^{-1} \circ a \circ a^{-1}$ Definition of Identity

Thus the criteria of an inverse semigroup are fulfilled.

$\blacksquare$