Elements in Same Left Coset iff Product with Inverse in Subgroup

Theorem

Let $G$ be a group

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

Let $x, y \in G$.

Then:

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

Proof

Let: $x H$ denote the left coset of $H$ by $x$

First we note that, from Congruence Class Modulo Subgroup is Coset, we have that the left 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:

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

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

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

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

$\Box$

Necessary Condition

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

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

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

Again, it follows from Congruence Class Modulo Subgroup is Coset that:

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

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

$\blacksquare$

Also see

• Elements in Same Right Coset iff Product with Inverse in Subgroup

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 42.6 \ \text {(4L)}$ Another approach to cosets