Elements in Same Left 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 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 $H$.

$\blacksquare$


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