Coset - Examples

Examples

Let G be the multiplicative group of {-1,1}, and H the trivial subgroup (1,*). Then -1H={-1}, 1H=H are the sole cosets of H in G.

Let G be the additive group of 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

are called affine subspaces, and are cosets (both left and right, since the group is abelian). In terms of geometric vectors, these affine subspaces are all the "lines" or "planes" parallel to the subspace, which is a line or plane going through the origin.