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 $\circ$

if and only if:

$\struct {H, \circ}$ 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

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