Elements in Same Right Coset iff Product with Inverse in Subgroup

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Theorem

Let $G$ be a group.

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

Let $x, y \in G$.


Then:

$x, y$ are in the same right coset of $H$ if and only if $x y^{-1} \in H$


Proof

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

First we note that, from Congruence Class Modulo Subgroup is Coset, we have that the right cosets of $G$ form a partition of $G$.


Sufficient Condition

Suppose $x, y$ are in the same left coset of $H$.

It follows from Congruence Class Modulo Subgroup is Coset that:

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

From Cosets are Equal iff Product with Inverse in Subgroup, we have that:

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

So if $x, y$ are in the same right coset of $H$ then $x y^{-1} \in H$.

$\Box$


Necessary Condition

Now suppose that $x y^{-1} \in H$.

From Right Cosets are Equal iff Product with Inverse in Subgroup, we have that:

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

It follows from Congruence Class Modulo Subgroup is Coset that:

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

and so:

$x, y$ are in the same left coset of $H$

Hence $x, y$ are in the same right coset of $H$ if $x y^{-1} \in H$.

$\blacksquare$


Also see


Sources

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