Right Cosets are Equal iff Element in Other Right Coset

Theorem

Let $G$ be a group.

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

Let $x, y \in G$.

Let $H x$ denote the right coset of $H$ by $x$.

Then:

$H x = H y \iff x \in H y$

Proof

\(\ds H x\)

\(=\)

\(\ds H y\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x y^{-1}\)

\(\in\)

\(\ds H\)

Right Cosets are Equal iff Product with Inverse in Subgroup

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(\in\)

\(\ds H y\)

Element in Right Coset iff Product with Inverse in Subgroup

$\blacksquare$

Also see

• Left Cosets are Equal iff Element in Other Left Coset

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Exercise $3$