Coset
In mathematics, specifically group theory, a subgroup H of a group G may be used to decompose the underlying set of G into disjoint, equal-size subsets called cosets.
Cosets also appear in other areas of mathematics such as vector spaces and error-correcting codes.
However, some authors (including Dummit & Foote and Rotman) reserve this notation specifically for representing the quotient group formed from the cosets in the case where H is a normal subgroup of G. Let G be the dihedral group of order six.
This is enough information to fill in the entire Cayley table: Let T be the subgroup {I, b}.
[2] Two elements being in the same left coset also provide a natural equivalence relation.
There are other types of equivalence relations in a group, such as conjugacy, that form different classes which do not have the properties discussed here.
Every left or right coset of H has the same number of elements (or cardinality in the case of an infinite H) as H itself.
Lagrange's theorem allows us to compute the index in the case where G and H are finite:
[5] (The same argument shows that every subgroup of an Abelian group is normal.
are called affine subspaces, and are cosets (both left and right, since the group is abelian).
If P is in R2, then the coset P + m is a line m′ parallel to m and passing through P.[8] Let G be the multiplicative group of matrices,[9]
That is, the left cosets consist of all the matrices in G having the same upper-left entry.
is not normal in G. A subgroup H of a group G can be used to define an action of H on G in two natural ways.
The orbit of g under the right action is the left coset gH, while the orbit under the left action is the right coset Hg.
[10] The concept of a coset dates back to Galois's work of 1830–31.
The term "co-set" apparently appears for the first time in 1910 in a paper by G. A. Miller in the Quarterly Journal of Pure and Applied Mathematics (vol.
Various other terms have been used including the German Nebengruppen (Weber) and conjugate group (Burnside).
Galois was concerned with deciding when a given polynomial equation was solvable by radicals.
A tool that he developed was in noting that a subgroup H of a group of permutations G induced two decompositions of G (what we now call left and right cosets).
If these decompositions coincided, that is, if the left cosets are the same as the right cosets, then there was a way to reduce the problem to one of working over H instead of G. Camille Jordan in his commentaries on Galois's work in 1865 and 1869 elaborated on these ideas and defined normal subgroups as we have above, although he did not use this term.
For instance, Hall (1959) would call gH a right coset, emphasizing the subgroup being on the right.
When a codeword (element of C) is transmitted some of its bits may be altered in the process and the task of the receiver is to determine the most likely codeword that the corrupted received word could have started out as.
This procedure is called decoding and if only a few errors are made in transmission it can be done effectively with only a very few mistakes.
A standard array is a coset decomposition of V put into tabular form in a certain way.
This element is called a coset leader and there may be some choice in selecting it.
An example of a standard array for the 2-dimensional code C = {00000, 01101, 10110, 11011} in the 5-dimensional space V (with 32 vectors) is as follows: The decoding procedure is to find the received word in the table and then add to it the coset leader of the row it is in.
Since in binary arithmetic adding is the same operation as subtracting, this always results in an element of C. In the event that the transmission errors occurred in precisely the non-zero positions of the coset leader the result will be the right codeword.
It is a method of computing the correct coset (row) that a received word will be in.
To decode, the search is now reduced to finding the coset leader that has the same syndrome as the received word.
[14] Two double cosets HxK and HyK are either disjoint or identical.
