Double coset

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 while $K$ acts 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

The set of all double cosets is denoted

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 $H \times K$ acting on $G$ by $(h, k)\cdot x = hxk -1$ . 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.

There is an equivalent description of double cosets in terms of single cosets. Let $H$ and $K$ both act by right multiplication on $G$ . Then $G$ acts by left multiplication on the product of coset spaces $G / H \times G / K$ . The orbits of this action are in one-to-one correspondence with $H \ G / K$ . This correspondence identifies $(xH, yK)$ with the double coset $Hx -1 yK$ . Briefly, this is because every $G$ -orbit admits representatives of the form $(H, xK)$ , and the representative $x$ is determined only up to left multiplication by an element of $H$ . Similarly, $G$ acts by right multiplication on $H \ G × K \ G$ , and the orbits of this action are in one-to-one correspondence with the double cosets $H \ G / K$ . Conceptually, this identifies the double coset space $H \ G / K$ with the space of relative configurations of an $H$ -coset and a $K$ -coset. Additionally, this construction generalizes to the case of any number of subgroups. Given subgroups $H 1, ..., H n$ , the space of $(H 1, ..., H n)$ -multicosets is the set of $G$ -orbits of $G / H 1 \times ... \times G / H n$ .

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