Double coset explained

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, generalizing the notion of a single coset.

Definition

Let be a group, and let and be subgroups. Let act on by left multiplication and let act on by right multiplication. For each in, the -double coset of is the set

HxK = \{ hxk \colon h \in H, k \in K \}. When, this is called the -double coset of . Equivalently, is the equivalence class of under the equivalence relation

if and only if there exist in and in such that .The set of all

(H,K) -double cosets is denoted by H \backslash G / K.

Properties

Suppose that is a group with subgroups and acting by left and right multiplication, respectively. The -double cosets of may be equivalently described as orbits for the product group acting on by . Many of the basic properties of double cosets follow immediately from the fact that they are orbits. However, because is a group and and 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 and are either disjoint or identical.
• is the disjoint union of its double cosets.
• There is a one-to-one correspondence between the two double coset spaces and given by identifying with .
• If