Coset

In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then If the group operation is written additively, the notation used changes to g + H or H + g. Cosets are a basic tool in the study of groups; for example they play a central role in Lagrange's theorem that states that for any finite group G, the number of elements of every subgroup H of G divides the number of elements of G. Although derived from a subgroup, cosets are not usually themselves subgroups of G. For example, it is not necessary for a coset to contain the identity element. If G is an abelian group, than gH = Hg for every subgroup H of G and every element g of G. In general, given an element g and a subgroup H of a group G, the right coset of H with respect to g is also the left coset of the conjugate subgroup g−1Hg  with respect to g, that is, Hg = g ( g−1Hg ). The number of left cosets of H in G is equal to the number of right cosets of H in G. The common value is called the index of H in G. A subgroup N of a group G is a normal subgroup of G if and only if for all elements g of G the corresponding left and right coset are equal, that is, gN = Ng. Furthermore, the cosets of N in G form a group called the quotient group or factor group. Let G = ({−1,1}, ×) be the group formed by {−1,1} under multiplication, which is isomorphic to C2, and H the trivial subgroup ({1}, ×). Then {−1} = (−1)H = H(−1) and {1} = 1H = H1 are the only cosets of H in G. Because its left and right cosets with respect to any element of G coincide, H is a normal subgroup of G. Let G be the additive group of the integers, Z = ({..., −2, −1, 0, 1, 2, ...}, +) and H the subgroup (mZ, +) = ({..., −2m, −m, 0, m, 2m, ...}, +) where m is a positive integer. Then the cosets of H in G are the m sets mZ, mZ + 1, ..., mZ + (m − 1), where mZ + a = {..., −2m+a, −m+a, a, m+a, 2m+a, ...}. There are no more than m cosets, because mZ + m = m(Z + 1) = mZ. The coset (mZ + a, +) is the congruence class of a modulo m. Another example of a coset comes from the theory of vector spaces. The elements (vectors) of a vector space form an abelian group under vector addition. It is not hard to show that subspaces of a vector space are subgroups of this group. For a vector space V, a subspace W, and a fixed vector a in V, the sets