Coset

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In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then

Only when H is normal will the right and left cosets of H coincide, which is one definition of normality of a subgroup.

A coset is a left or right coset of some subgroup in G. Since Hg = g ( g−1Hg ), the right cosets Hg (of H ) and the left cosets g ( g−1Hg ) (of the conjugate subgroup g−1Hg ) are the same. Hence it is not meaningful to speak of a coset as being left or right unless one first specifies the underlying subgroup. In other words: a right coset of one subgroup equals a left coset of a different (conjugate) subgroup. Whether a coset is a right coset or a left coset depends on which subgroup is used.

For abelian groups or groups written additively, the notation used changes to g+H and H+g respectively.

Contents

Examples

The additive cyclic group Z4 = {0, 1, 2, 3} = G has a subgroup H = {0, 2} (isomorphic to Z2). The left cosets of H in G are

There are therefore two distinct cosets, H itself, and 1 + H = 3 + H. Note that every element of G is either in H or in 1 + H, that is, H ∪ (1 + H ) = G, so the distinct cosets of H in G partition G. Since Z4 is an abelian group, the right cosets will be the same as the left.

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.

General properties

gH is an element of H if and only if g is an element of H, since as H is a subgroup, it must be closed and must contain the identity.

Any two left cosets of H in G are either identical or disjoint  — i.e., the left cosets form a partition of G such that every element of G belongs to one and only one left coset.[1] In particular the identity is in precisely one coset, and that coset is H itself; this is also the only coset that is a subgroup. We can see this clearly in the above examples.

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