Left and Right Coset Spaces are Equivalent

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

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

Let:
 * $x H$ denote the left coset of $H$ by $x$
 * $H y$ denote the right coset of $H$ by $y$.

Then:
 * $\order {\set {x H: x \in G} } = \order {\set {H y: y \in G} }$

That is:
 * The number of right cosets is the same as the number of left cosets of $G$ with respect to $H$.


 * The left and right coset spaces are equivalent.

Also see

 * Index of Subgroup, which is the number of left (or right) cosets of a subgroup.


 * Subgroup is Normal iff Left Cosets are Right Cosets