# Definition:Normal Subgroup of Monoid

## Definition

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

Let $\struct {H, \circ}$ be a subgroup of $\struct {S, \circ}$.

$H$ is a normal subgroup of $S$ if and only if:

$e$ is the identity element of $H$
$\forall s \in S: s \circ H = H \circ s$

where $s \circ H$ denotes the subset product of $s$ with $H$.

## Also defined as

It will be noted that this is the same definition as a normal subgroup of a group, which is the usual context in which to find this definition.

## Also see

• Results about normal subgroups can be found here.