Double coset of a pair of subgroups

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Definition

Definition with symbols

Let H and K be subgroups of a group G. Then a subset L of G is termed a double coset for H and K if the following equivalent conditions are satisfied:

  • There exists an element x in G such that L = HxK
  • For any element x in L, L = HxK.

Facts

Equivalence relation

The double cosets of a pair of subgroups are pairwise disjoint and hence form a partition of the group. The relation of being in the same double coset is an equivalence relation on the elements of the group.

Special cases

Let M be a subgroup of G. We can consider the following three special cases:

  • H is trivial and K = M. In this case, the double cosets of H and K are the same as the left cosets of M
  • H = K = M. In this case, the double cosets of H and K are simply called the double cosets of M.

For a normal subgroup

For a normal subgroup, the notions of left coset, right coset, and double coset are equivalent.

Double coset index

The double coset index of a pair of subgroups is the number of double cosets.

The double coset index of a subgroup is the number of double cosets it has as a subgroup (that is, where both subgroups are equal to the given subgroup).

Note that the double coset index equals the usual index if and only if the subgroup is normal.

Bound on double coset index in terms of orders of group and subgroup.

Double coset cardinality

The cardinality \left| HxK \right| of the double coset HxK is \left| H \right| \left| K \right| / \left| H \cap xKx^{-1} \right| .

HxK is the union of 
    [H:H \cap xKx^{-1}]
  = \left| HxK \right| / \left| K \right|
different (and hence disjoint) left cosets of K.


    [G:K] = \sum_{x \in X} [H:H \cap xKx^{-1}]
where X is a set of representatives of the double cosets of H and K.

Quasiorder on a collection of double cosets

We can order the double cosets of a pair of subgroups by the translate-containment quasiorder, wherein we say HxK \le HyK if there exist g,h \in G such that gHxKh \subseteq HyK.

Under this quasiorder, all the double cosets that actually comprise single cosets are the smallest or minimal elements -- these are in fact precisely the single cosets inside the normalizer of the subgroup. A pair of subgroups for which this quasiorder is actually a partial order (that is, two distinct double cosets cannot both be translate-contained in the other) is termed a double coset-ordering subgroup pair. When both member subgroups are the same, we call it a double coset-ordering subgroup.

Double coset space in terms of orbits under a group action

Consider the product of the left coset spaces G/H and G/K. G acts on both these coset spaces by left multiplication, and we can hence consider the action of G on the product of the coset spaces G/H \times G/K. The orbits under this action are the double cosets of H and K in G.

The proof of this comes from the fact that for every orbit, we can choose a representative where the first coordinate is the coset H itself. In this case, the second representative gives a coset of K. However, this coset is ambiguous upto left multiplication by an element of H. So the upshot is that we get a union of left cosets of K, under the left action of H. This is a double coset of H and K.

Double cosets thus measure the relative position of two left cosets.

This alternative approach to double cosets also allows us to generalize to the notion of a multicoset for a tuple of subgroups.