Element in Right Coset iff Product with Inverse in Subgroup

Theorem

Let $G$ be a group.

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

Let $x, y \in G$.

Let $H \circ y$ denote the right coset of $H$ by $y$.

Then:

$x \in H y \iff x y^{-1} \in H$

Proof

Let $\struct {G, *}$ be the opposite group of $G$.

Then:

$x \in H y \iff x \in y * H$

$x y^{-1} \in H \iff y^{-1} * x \in H$

Since $H$ is closed under inverses:

$x y^{-1} \in H \iff x^{-1} * y \in H$

By Element in Left Coset iff Product with Inverse in Subgroup:

$x \in y * H \iff x^{-1} * y \in H$

Hence the result.

$\blacksquare$

Also see

• Element in Left Coset iff Product with Inverse in Subgroup

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.1$. The quotient sets of a subgroup: Lemma $\text{(i)}$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 42.6 \ \text {(1R)}$ Another approach to cosets