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 $\struct {S, \circ}$.
 * $\struct {H, \circ}$ is a normal subgroup of $\struct {S, \circ}$.

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 $\circ$.