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 $a,b \in G$ , we define

$\begin{displaymath} a \sim b \ \ \ \ {\rm if\ and\ only\ if}\ \ \ h_1ah_2 = b \end{displaymath}$

(8.3)

where $h_1 \in H_1$ and $h_2 \in H_2$ . We contend first of all that the relation given in (8.3) is an equivalence relation on

. To see this, we note

(Reflexivity): $a \sim a$ since , where is the identity element and $e \in H_1$ and $e \in H_2$ since $H_1\leq G$ , $H_2 \leq G$ .
(Symmetry): $a \sim b$ implies there exist $h_1 \in H_1$ and $h_2 \in H_2$ such that , but then $a=h_1^{-1}bh_2^{-1}$ , so $b \sim a$ since $h_1^{-1} \in H_1$ and $h_2^{-1} \in H_2$ .
(Transitivity): If $a \sim b$ and $b \sim c$ , then there exist elements $h_1,h_1'\in H_1$ and $h_2,h_2'\in H_2$ such that

$\begin{displaymath} h_1ah_2=b\ \ \ \ {\rm and}\ \ \ h_1'ah_2'=c, \end{displaymath}$

whence

$\begin{displaymath} h_1'h_1ah_2'h_2=c. \end{displaymath}$

Since $h_1h_1'\in H_1$ and $h_2h_2'\in H_2$ , we have that $a \sim c$ .

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

Definition 8.2.1 Let be a group with subgroups and (not necessarily distinct). If $a \in G$ , the complex is called a double coset with respect to and . By definition,

$\begin{displaymath} H_1aH_2 = \{h_1ah_2 \in G \ \vert\ h_1 \in H_1, h_2 \in H_2\}. \end{displaymath}$

For $a \in G$ , the equivalence class of as we recall, contains all $b \in G$ with $b \sim a$ . By the definition given in (8.3), this means , where $h_1 \in H_1$ and $h_2 \in H_2$ . Thus $b \in H_1aH_2$ . 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

$\begin{displaymath} H_1aH_2 = H_1bH_2\ \ \ {\rm or}\ \ \ H_1aH_2 \cap H_1bH_2 = \emptyset \end{displaymath}$

and

$\begin{displaymath} G =\coprod_a H_1aH_2 \end{displaymath}$

where the union is taken over certain $a \in G$ . The identity element

belongs to the complex

If $H_2 = \{e\}$ , then we simply get the right coset decomposition of with respect to . If $H_1 = \{e\}$ , 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 $h_2 \in H_2$ and contains all left cosets of the form where $h_1 \in H_1$ . We claim, as a matter of fact, that is a union of right or left cosets of the above form. For suppose that

$\begin{displaymath} gH_2 \cap H_1aH_2\not= \emptyset. \end{displaymath}$

Then there exist elements $h_2,h_2'\in H_2$ and $h_1 \in H_1$ such that

$\begin{displaymath} gh_2'=h_1ah_2 \end{displaymath}$

or $g=h_1ah_2(h_2')^{-1}$ . This implies that

$\begin{displaymath} gH_2=h_1aH_2, \end{displaymath}$

and so $gH_2 \subset H_1aH_2$ . Since this shows that any left coset which has anything at all in common with

, must be totally contained in

, we have

$\begin{displaymath} H_1aH_2 = \cup_{h_1} h_1aH_2. \end{displaymath}$

(8.4)

Similarly, it can be shown that

$\begin{displaymath} H_1aH_2 = \cup_{h_2} H_1ah_2 . \end{displaymath}$

(8.5)

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 $\vert G\vert < \infty$ . This is contained in

Theorem 8.2.2 Let be a finite group, let $H_1\leq G$ , $H_2 \leq G$ , and let $a \in G$ . Then

(a): The number of right cosets of in is $[H_2 : H_2 \cap a^{-1}H_1a]$ .
(b): The number of left cosets of in is $[a^{-1}H_1a : H_2 \cap a^{-1}H_1a]$ .

Proof: We first note that $a^{-1}H_1a$ is a subgroup by Proposition 7.2.1. Consider the mapping of the double coset onto the complex $a^{-1}H_1aH_2$ given by $h_1ah_2\longmapsto a^{-1}h_1ah_2$ . It is easy to show that this map is well-defined, 1-1, and onto (see exercise 2 for this section). Thus $\vert H_1aH_2\vert = \vert a^{-1}H_1aH_2\vert$ . But $a^{-1}H_1aH_2$ is the product of two subgroups $a^{-1}H_1a$ and so by the product theorem (Theorem 4.3.6),

$\begin{displaymath} \vert a^{-1}H_1a\cdot H_2 \vert = \frac{\vert a^{-1}H_1a\vert\vert H_2 \vert}{\vert a^{-1}H_1a\cap H_2 \vert}. \end{displaymath}$

Now according to (8.4), the number of left cosets of

is $\vert H_1aH_2\vert/ \vert H_2\vert$ . Thus the number of left cosets of

$\begin{displaymath} \frac{\vert a^{-1}H_1a\vert}{\vert a^{-1}H_1a\cap H_2 \vert} =[a^{-1}H_1a:a^{-1}H_1a\cdot H_2 ]. \end{displaymath}$

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

Under the same hypotheses as in Theorem 8.2.2, we use the notation $\char93 (H_1aH_2)$ to be the number of right cosets of in times $\vert H_1\vert$ (note from (8.5) that $\char93 (H_1aH_2) = \vert H_1aH_2\vert$ ). Thus Theorem 8.2.2 implies that

$\begin{displaymath} \char93 (H_1aH_2)= \frac{\vert H_2\vert}{\vert a^{-1}H_1a\cap H_2 \vert}\cdot \vert H_1\vert. \end{displaymath}$

This proves the following result.

Corollary 8.2.3 Let be a finite group and let $H_1\leq G$ , $H_2 \leq G$ . If $G=\coprod_a H_1aH_2$ (disjoint), then

$\begin{displaymath} \vert G\vert=\sum_{j=1}^n \frac{\vert H_1\vert\vert H_2\vert}{d_j} \end{displaymath}$

(8.6)

where $d_j=\vert a_j^{-1}H_1a_j\cap H_2 \vert$ .

Subsections

Exercises

David Joyner 2007-08-06