Coset

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

gH = { gh : h an element of H } is a left coset of H in G, and

Hg = { hg : h an element of H } is a right coset of H in G.

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

Examples

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

0 + H = {0, 2} = H

1 + H = {1, 3}

2 + H = {2, 0} = H

3 + H = {3, 1}

From above, it is clear that there are two distinct cosets, H itself, and 1+H = 3 + H. Note that H ∪ 1+H = G, so the different cosets of H in G partition G. Since Z₄ is an abelian group, the right cosets will be the same as the left (this is not difficult to verify).

Another famous example of a coset comes in the theory of vector spaces. 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. Now for a vector space V, a subspace W, and a fixed vector a in V, the sets

\{x \in V \colon x = a + n, n \in W\}

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

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

Any two left cosets are either identical or disjoint -- the left cosets form a partition of G: every element of G belongs to one and only one left coset. In particular the identity is only in one coset, and H itself is the only coset that is a subgroup. We can see this clearly in the above example. The left cosets of H in G are the equivalence classes under the equivalence relation on G given by x ~ y if and only if x^-1y ∈ H. Similar statements are also true for right cosets.

A coset representative is a representative in the equivalence class sense. A set of representatives of all the cosets is called a transversal. 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. Some books on very applied group theory erroneously identify the conjugacy class as 'the' equivalence class as opposed to a particular type of equivalence class.

All left cosets and all right cosets have the same number of elements (or cardinality in the case of an infinite H). Furthermore, the number of left cosets is equal to the number of right cosets and is known as the index of H in G, written as : H. Lagrange's theorem allows us to compute the index in the case where G and H are finite, as per the formula:

|G| = : H · |H|

This equation also holds in the case where the groups are infinite (but is somewhat less useful).

The subgroup H is normal if and only if gH = Hg for all g in G. In this case one can turn the set of all cosets into a group, the factor group of G by H.

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Examples

General properties

See also