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Concept# Double coset

Summary

In group theory, a field of mathematics, a double coset is a collection of group elements which are equivalent under the symmetries coming from two subgroups. More precisely, let G be a group, and let H and K be subgroups. Let H act on G by left multiplication and let K act on G by right multiplication. For each x in G, the (H, K)-double coset of x is the set
When H = K, this is called the H-double coset of x. Equivalently, HxK is the equivalence class of x under the equivalence relation
x ~ y if and only if there exist h in H and k in K such that hxk = y.
The set of all -double cosets is denoted by
Suppose that G is a group with subgroups H and K acting by left and right multiplication, respectively. The (H, K)-double cosets of G may be equivalently described as orbits for the product group H × K acting on G by (h, k) ⋅ x = hxk−1. Many of the basic properties of double cosets follow immediately from the fact that they are orbits. However, because G is a group and H and K are subgroups acting by multiplication, double cosets are more structured than orbits of arbitrary group actions, and they have additional properties that are false for more general actions.
Two double cosets HxK and HyK are either disjoint or identical.
G is the disjoint union of its double cosets.
There is a one-to-one correspondence between the two double coset spaces H \ G / K and K \ G / H given by identifying HxK with Kx−1H.
If H = {1}, then H \ G / K = G / K. If K = {1}, then H \ G / K = H \ G.
A double coset HxK is a union of right cosets of H and left cosets of K; specifically,
The set of (H, K)-double cosets is in bijection with the orbits H \ (G / K), and also with the orbits (H \ G) / K under the mappings and respectively.
If H is normal, then H \ G is a group, and the right action of K on this group factors through the right action of H \ HK. It follows that H \ G / K = G / HK. Similarly, if K is normal, then H \ G / K = HK \ G.
If H is a normal subgroup of G, then the H-double cosets are in one-to-one correspondence with the left (and right) H-cosets.

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