# 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.

### C

### E

### G

### L

## Pages in category "Cosets"

The following 40 pages are in this category, out of 40 total.

### C

- Condition for Group to Act Effectively on Left Coset Space
- Congruence Class Modulo Subgroup is Coset
- Conjugates of Elements in Centralizer
- Coset by Identity
- Coset Equals Subgroup iff Element in Subgroup
- Coset of Subgroup of Subgroup
- Coset of Trivial Subgroup is Singleton
- Coset Space forms Partition
- Cosets are Equal iff Element in Other Coset
- Cosets are Equal iff Product with Inverse in Subgroup
- Cosets are Equivalent
- Cosets in Abelian Group

### E

- Element in Coset iff Product with Inverse in Subgroup
- Element in Left Coset iff Product with Inverse in Subgroup
- Element in Right Coset iff Product with Inverse in Subgroup
- Element of Group is in its own Coset
- Element of Group is in Unique Coset of Subgroup
- Elements in Same Coset iff Product with Inverse in Subgroup
- Elements in Same Left Coset iff Product with Inverse in Subgroup
- Elements in Same Right Coset iff Product with Inverse in Subgroup

### L

- Left and Right Coset Spaces are Equivalent
- Left Congruence Class Modulo Subgroup is Left Coset
- Left Coset Equals Subgroup iff Element in Subgroup
- Left Coset of Stabilizer in Group of Transformations
- Left Cosets are Equal iff Element in Other Left Coset
- Left Cosets are Equal iff Product with Inverse in Subgroup

### R

- Regular Representation on Subgroup is Bijection to Coset
- Right Congruence Class Modulo Subgroup is Right Coset
- Right Coset Equals Subgroup iff Element in Subgroup
- Right Cosets are Equal iff Element in Other Right Coset
- Right Cosets are Equal iff Left Cosets by Inverse are Equal
- Right Cosets are Equal iff Product with Inverse in Subgroup