Element of Group is in Unique Coset of Subgroup/Right

Theorem

Let $G$ be a group.

Let $H$ be a subgroup of $G$.

Let $x \in G$.

There exists a exactly one right coset of $H$ containing $x$, that is: $H x$

Proof

Follows directly from:

• Right Congruence Modulo Subgroup is Equivalence Relation
• Element in its own Equivalence Class.

$\blacksquare$

Also see

• Element of Group is in Unique Left Coset of Subgroup

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 42.5$ Another approach to cosets