TITLE: Motivation and intuition of double cosets
QUESTION [7 upvotes]: 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?

REPLY [3 votes]: 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.