Element in 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 $x, y \in G$.


Let $y H$ denote the left coset of $H$ by $y$.


Then:

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


Proof

\(\displaystyle x\) \(\in\) \(\displaystyle y H\)
\(\displaystyle \iff \ \ \) \(\displaystyle \exists h \in H: \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle y h\) Definition of Left Coset
\(\displaystyle \iff \ \ \) \(\displaystyle \exists h \in H: \ \ \) \(\displaystyle x^{-1}\) \(=\) \(\displaystyle h^{-1} y^{-1}\) Inverse of Group Product
\(\displaystyle \iff \ \ \) \(\displaystyle \exists h \in H: \ \ \) \(\displaystyle x^{-1} y\) \(=\) \(\displaystyle h^{-1}\) Product with $y$ on the right
\(\displaystyle \iff \ \ \) \(\displaystyle x^{-1} y\) \(\in\) \(\displaystyle H\) $H$ is a subgroup

$\blacksquare$


Also see


Sources