Element in Coset iff Product with Inverse in Subgroup

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\struct {G, \circ}$ be a group.

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

Let $x, y \in G$.


Element in Left Coset iff Product with Inverse in Subgroup

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


Then:

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


Element in Right Coset iff Product with Inverse in Subgroup

Let $H \circ y$ denote the right coset of $H$ by $y$.


Then:

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