Double cosets

We turn now to another important decomposition of an arbitrary group G into disjoint complexes. Such decompositions will play an important role in our later considerations of the Sylow Theorems.

Let be an arbitrary group and let and be
subgroups of . For , we define

- (Reflexivity): since , where is
the identity element and and
since , .
- (Symmetry): implies there exist
and such that , but then
, so
since
and
.
- (Transitivity): If and ,
then there exist elements
and
such that

whence

Since and , we have that .

Thus given by (8.3) is indeed an equivalence relation on . We next take a look at the equivalence classes.

For , the equivalence class of
as we recall, contains all with
.
By the definition given in (8.3),
this means ,
where and .
Thus
. As the above statements are
all ``if and only if'',
we see that , the double coset given in
Definition 8.2.1.
By our general theorem on equivalence
relations, Theorem 1.1.4,
we know that either

and

where the union is taken over certain . The identity element belongs to the complex .

If , then we simply get the right coset decomposition of with respect to . If , then we have the left coset decomposition of with respect to . Thus the double coset decomposition of a group may be viewed as a generalization of the coset (right or left) decomposition of a group. However, the reader should be careful not to generalize all facts related to coset decompositions to the case of double coset decompositions. For example, we saw that any two cosets of a finite group have the same number of elements. We shall presently see that this is not the case with double cosets.

Let us consider the double coset .
Clearly contains all right cosets of the
form , where and
contains all left cosets of the
form where .
We claim, as a matter of fact, that
is a union of right or left cosets of
the above form. For suppose that

Then there exist elements and such that

or . This implies that

and so . Since this shows that any left coset which has anything at all in common with , must be totally contained in , we have

Similarly, it can be shown that

Next, we wish to ascertain the number of left and right cosets in the double coset. Even though this number can be finite for an infinite double coset, we assume . This is contained in

- (a)
- The number of right cosets of
in is
.
- (b)
- The number of left cosets of in is .

**Proof:**
We first note that is a subgroup by
Proposition 7.2.1. Consider the mapping of the double
coset onto the complex
given by
.
It is easy to show that this map is well-defined, 1-1,
and onto (see exercise 2 for this section).
Thus
. But is the
product of two subgroups and so by the
product theorem (Theorem 4.3.6),

Now according to (8.4), the number of left cosets of in is . Thus the number of left cosets of in is

This establishes part (b) of the theorem. A similar argument establishes part (a) (this is left as an exercise).

Under the same hypotheses as in Theorem 8.2.2, we use
the notation to be the number of right
cosets of in times
(note from (8.5) that
).
Thus Theorem 8.2.2 implies that

This proves the following result.

David Joyner 2007-08-06