# Condition for Subgroup of Monoid to be Normal

## Theorem

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

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

Then:

the set of left cosets of $\struct {H, \circ}$ in $\struct {S, \circ}$ form a partition of $S$
and:
the set of right cosets of $\struct {H, \circ}$ in $\struct {S, \circ}$ form a partition of $S$
such that the equivalence relations induced by those partitions are congruence relations for $\circ$
$\struct {H, \circ}$ is a normal subgroup of $\struct {S, \circ}$.

## Proof

### Necessary Condition

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

Then by definition:

$e \in H$

Hence from Condition for Cosets of Subgroup of Monoid to be Partition, the set of left cosets and the set of right cosets of $\struct {H, \circ}$ in $\struct {S, \circ}$ form a partition of $S$.

### Sufficient Condition

Let $\struct {H, \circ}$ be a subgroup of $\struct {S, \circ}$ such that:

the set of left cosets of $\struct {H, \circ}$ in $\struct {S, \circ}$ form a partition of $S$

and:

the set of right cosets of $\struct {H, \circ}$ in $\struct {S, \circ}$ form a partition of $S$

such that the equivalence relations induced by those partitions are congruence relations for $\circ$.