# Condition for Subgroup of Monoid to be Normal

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## 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 $S$.

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### 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 $\circ$.

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.16$