# Category:Cosets

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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:

- $x \circ H = \set {y \in S: \exists h \in H: y = x \circ h}$

That is, it is the subset product with singleton:

- $x \circ H = \set x \circ H$

### Right Coset

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

- $H \circ y = \set {x \in S: \exists h \in H: x = h \circ y}$

That is, it is the subset product with singleton:

- $H \circ y = H \circ \set y$

## Subcategories

This category has the following 11 subcategories, out of 11 total.

### C

- Coset by Identity (3 P)
- Coset Space forms Partition (5 P)
- Cosets are Equivalent (3 P)

### E

- Examples of Cosets (20 P)

### G

- Group Action on Coset Space (6 P)

### L

## Pages in category "Cosets"

The following 41 pages are in this category, out of 41 total.

### C

- Condition for Cosets of Subgroup of Monoid to be Partition
- 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