Right Cosets are Equal iff Product with Inverse in Subgroup

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $G$ be a group.

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

Let $x, y \in G$.

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


Then:

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


Proof

\(\displaystyle H x\) \(=\) \(\displaystyle H y\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x\) \(\equiv^r\) \(\displaystyle y \bmod H\) Right Coset Space forms Partition
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x y^{-1}\) \(\in\) \(\displaystyle H\) Equivalent Statements for Congruence Modulo Subgroup

$\blacksquare$


Also see


Sources