Right Coset Equals Subgroup iff Element in Subgroup

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $G$ be a group whose identity is $e$.

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

Let $x \in G$.

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


Then:

$H x = H \iff x \in H$


Proof

\(\ds H x\) \(=\) \(\ds H\)
\(\ds \leadstoandfrom \ \ \) \(\ds H x\) \(=\) \(\ds H e\) Right Coset by Identity: $H = H e$
\(\ds \leadstoandfrom \ \ \) \(\ds x e^{-1}\) \(\in\) \(\ds H\) Right Cosets are Equal iff Product with Inverse in Subgroup
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(\in\) \(\ds H\) Group Properties

$\blacksquare$


Also see


Sources