Q:

Motivation and intuition of double cosets

In Dummit and Foote, there was a problem on double cosets. Let $H$ and $K$ be subgroups of $G$ and $x\in G$, $HxK$ is the double coset of $x$. I can prove the double cosets partitions $G$. However, I could not see any intuition or motivation of this gear. I can understand it as a mapping $H:\frac{G}{K}\to gK$ or $K:Hx\to Hy$ where $y$ may be different from $x$. Is there anything deep for this motivation other than the mapping itself?

A:

The transitive $G$-sets are precisely those isomorphic to the coset space $G/H$ for some subgroup $H\le G$. Any of our $G$-sets decomposes as a disjoint union of orbits. So we should be able to decompose $G/H\times G/K$ into orbits. Indeed we have the decomposition
$$G/H\times G/K\cong\bigsqcup_{HxK}\frac{G}{H\cap {}^xK} \tag{$\circ$}$$
as $HxK$ ranges over all double cosets. (While the expression $H\cap{}^xK$ may be asymmetrical, we know that it's conjugate by $x$ to $H^x\cap K$, and $G/A\cong G/B$ if $A\sim B$ are conjugate.)
This would be useful for finding the structure constants of the Burnside rings. As all finite $G$-sets are disjoint unions of orbits, we can think of them as $\Bbb N$-linear sums of orbits, and then generalize to formal $\Bbb Z$-linear combinations of orbits. We can then extend the direct product operation, originally only defined for $\Bbb N$-linear combinations, to all these formal linear combinations and this creates a multiplication operation. This (in general we call this a Grothendieck construction) is how we form the Burnside ring.
There is an analogous situation with the complex linear representations of the group. All finite-dimensional representations decompose as direct sums of irreducible representations. We have direct sum and tensor product operations, and when we allow formal $\Bbb Z$-linear combinations of irreducible representations we get the Green ring, also known as the representation ring.
In quantum physics and Lie algebra theory, these structure constants are called Clebsch-Gordan coefficients. While in the context of combinatorics, symmetric polynomials and the representation theory of the symmetric group (all inextricably linked), the structure constants of ${\rm Rep}(G)$ with $G=S_n$ a symmetric group are called the Littlewood-Richardson coefficients. So we see that the structure constants of the Burnside ring are at least analogous to some pretty important invariants in advanced mathematics, and double cosets are related to them by $(\circ)$.
(It's also possible that my answer is a bit ... reaching.)

A:

The double coset arises in representation theory of finite groups, in particular Mackey's Decomposition Theorem (a very important and useful result!). 

A:

Double cosets can be used to describe the subgroups of free products and free products with amalgamation (which are important structures in the theory of infinite groups. See the paper The subgroups of a free product of two groups with an amalgamated subgroup by Karrass and Solitar (1970). The description of the subgroups of a free product with amalgamation proven by Karrass and Solitar is as follows (restricting to free products, this is more complete than, and implies, the Kurosh subgroup theorem).
Theorem: Let $G = A \ast_U B$ be a free product with amalgamation. The structure of a
subgroup $H$ may be described as follows: there exist double coset representative systems $\{D_{\alpha}\}$, $\{D_{\beta}\}$ for $G \mod (H, A)$ and $G \mod (H, B)$ respectively,
and there exists a set of elements $t_1, t_2,\ldots$ (possibly empty) such that

$H=\langle t_1 t_2,\ldots, D_{\alpha}AD_{\alpha}^{-1} \cap H,\ldots, D_{\beta}AD_{\beta}^{-1} \cap H,\ldots\rangle$;
$t_1, t_2,\ldots$ freely generate a free group $F$;
$D_{\alpha}AD_{\alpha}^{-1} \cap H,\ldots, D_{\beta}AD_{\beta}^{-1} \cap H,\ldots$ generate a generalised free product $S$ with amalgamated subgroups $D_{\alpha}UD_{\alpha}^{-1} \cap H,\ldots, D_{\beta}UD_{\beta}^{-1} \cap H,\ldots$; and
$H$ is the free product of the groups $T_i=\langle t_i, S\rangle$ with the subgroups $S$ amalgamated.

