Group of units of a monoid

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Definition

In words

Given a monoid, its group of units is the set of all invertible elements in the monoid under the monoid's operation. It is indeed a group.

In symbols

Let $M$ be a monoid. Then $G = {m \in M : \exists n \in M : n m = e}$ . Then $G$ is a group, the group of units of a monoid.

Examples

One simple family of examples: The group of units modulo n is the group of units for the monoid of integers modulo n under multiplication.