coset
Sign in to savethumb| is the group \mathbb{Z}/8\mathbb{Z}, the Integers modulo n|integers mod 8 under addition. The subgroup contains only 0 and 4. There are four left cosets of : itself, , , and (written using additive notation since this is the [[additive group). Together they partition the entire group into equal-size, non-overlapping sets. The index is 4.]]
Article
20 sectionsContents
- Definition
- First example
- Properties
- Normal subgroups
- Index of a subgroup
- More examples
- Integers
- Vectors
- Matrices
- As orbits of a group action
- History
- An application from coding theory
- Double cosets
- Notation
- More applications
- See also
- Notes
- References
- Further reading
- External links
thumb| is the group \mathbb{Z}/8\mathbb{Z}, the Integers modulo n|integers mod 8 under addition. The subgroup contains only 0 and 4. There are four left cosets of : itself, , , and (written using additive notation since this is the [[additive group). Together they partition the entire group into equal-size, non-overlapping sets. The index is 4.]]
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are left cosets and right cosets. Cosets (both left and right) have the same number of elements (cardinality) as does . Furthermore, itself is both a left coset and a right coset. The number of left cosets of in is equal to the number of right cosets of in . This common value is called the index of in and is usually denoted by .