# Category:Cosets

This category contains results about Cosets.
Definitions specific to this category can be found in Definitions/Cosets.

### Left Coset

The left coset of $x$ modulo $H$, or left coset of $H$ by $x$, is:

$x H = \set {y \in G: \exists h \in H: y = x h}$

This is the equivalence class defined by left congruence modulo $H$.

That is, it is the subset product with singleton:

$x H = \set x H$

### Right Coset

The right coset of $y$ modulo $H$, or right coset of $H$ by $y$, is:

$H y = \set {x \in G: \exists h \in H: x = h y}$

This is the equivalence class defined by right congruence modulo $H$.

That is, it is the subset product with singleton:

$H y = H \set y$

## Subcategories

This category has the following 9 subcategories, out of 9 total.