Coset
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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.
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.
For abelian groups or groups written additively, the notation used changes to g+H and H+g respectively.
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[edit] 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
- 0 + H = {0, 2} = H
- 1 + H = {1, 3}
- 2 + H = {2, 0} = H
- 3 + H = {3, 1}
There are therefore two distinct cosets, H itself, and 1 + H = 3 + H. Note that 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 (this is not difficult to verify).
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.
[edit] General properties
We have gH = 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 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 that coset is H itself; this is also the only coset that is a subgroup. We can see this clearly in the above examples.
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 order (number of elements, or cardinality in the case of an infinite H), equal to the order of H (because H is itself a coset). 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 [G : H ]. Lagrange's theorem allows us to compute the index in the case where G and H are finite, as per the formula:
- |G | = [G : H ] · |H |
This equation also holds in the case where the groups are infinite, although the meaning may be less clear.
[edit] Cosets and normality
If H is not normal in G, then its left cosets are different from its right cosets. That is, there is an a in G such that no element b satisfies aH = Hb. This means that the partition of G into the left cosets of H is a different partition than the partition of G into right cosets of H. (It is important to note that some cosets may coincide. For example, if a is in the center of G, then aH = Ha.)
On the other hand, the subgroup N is normal if and only if gN = Ng for all g in G. In this case, the set of all cosets form a group called the quotient group G /H with the operation ∗ defined by (aH )∗(bH ) = abH. Since every right coset is a left coset, there is no need to differentiate "left cosets" from "right cosets".
[edit] Finite index
An infinite group G may have subgroups H of finite index (for example, the even integers inside the group of integers). Such a subgroup always contains a normal subgroup N (of G), also of finite index. In fact, if H has index n, then the index of N can be taken as some factor of n!. This can be seen more concretely, by considering the permutation action of G by multiplication on the left cosets of H (or, equally, on the right cosets). This provides a quotient group of G, the kernel of this permutation representation, which is a subgroup of the symmetric group on n elements.
A special case, n = 2, gives the general result that a subgroup of index 2 is a normal subgroup.
