This is taken from Lang's Algebra, exercises on groups.
Let $G$ be a group and $H,H'$ be subgroups of $G$. A double coset of $H,H'$ is a subset of the form $HxH'$, $x\in G$. The first question asks to prove that "$G$ is a disjoint union of double cosets."
Well, we have $G = \coprod_{g\in G} \{e\}g\{e\}$, but I guess this is not what Lang expected to see as an answer. So let us take two subgroups of $G$, $H$ and $H'$ and define the action $\rho: H\times H' \rightarrow \mathfrak S_G$ defined by $\rho(h,h') = (x \mapsto hxh'^{-1})$. Hence the orbit of an element $x$ is $HxH'$, and taking $X$ as a set of representatives of the orbits, $G = \coprod_{x\in X} HxH'$.
Is this the "expected (canonical) answer"? Are there other "canonical" decompositions into cosets?
Best Answer
Hints:
Prove that
$$R\subset G\times G\;,\;\;(x,y)\in R\iff \;\exists \;h\in H\;,\;h'\in H'\;\;s.t.\;\;x=hyh'$$
is an equivalence relation on $\,G\;$ , and thus $\;G\;$ is the disjoint union of this equivalence relation's equivalence classes.
Now, what are the equivalence classes? :)