Let $G$ be a finite abelian group, and let $A$ and $B$ be subgroups. I'm interested in $G/A\times G/B$ with its natural $G$-set structure.
In $G/A\times G/B$, the stabilizer of any element is $A\cap B$, so by the orbit-stabilizer theorem, there is a decomposition into orbits like this: $$G/A\times G/B \cong \coprod (G/A\cap B)$$ However, I would like to make this isomorphism of $G$-sets explicit - that is, I want to find a natural choice of elements $\{x_1,\ldots,x_k\}$ of $G/A\times G/B$ whose orbits are the copies of $G/A\cap B$: $$G/A\times G/B=Gx_1\sqcup \cdots \sqcup Gx_k,\quad\text{each }Gx_i\cong G/A\cap B\text{ as $G$-sets}$$ For example, suppose that $G$ is cyclic. Let's say $$G = \mathbb{Z}/n\mathbb{Z} = \{\overline{0},\overline{1},\ldots,\overline{n-1}\},\quad A=a\mathbb{Z}/n\mathbb{Z},\quad B=b\mathbb{Z}/n\mathbb{Z}$$ Then it's easy to see that $$k=\frac{[G:A][G:B]}{[G:A\cap B]} = \frac{ab}{\mathrm{lcm}(a,b)}=\gcd(a,b)$$ and that (by the Chinese remainder theorem) a good choice of representatives is $$x_1=(\overline{0}+A,\,\overline{0}+B)\;\;\ldots\;\;x_k=(\overline{0}+A,\,\overline{k-1}+B)$$ or, just as well, $$x_1=(\overline{0}+A,\,\overline{0}+B)\;\;\ldots\;\;x_k=(\overline{k-1}+A,\,\overline{0}+B)$$
But now if $G$ is an arbitrary finite abelian group $G=\mathbb{Z}/p_1^{a_1}\mathbb{Z}\times \cdots\times\mathbb{Z}/p_r^{a_r}\mathbb{Z}$, can we find a similarly explicit choice of representatives?
I tried to generalize the cyclic case and couldn't get anywhere. I'd appreciate some help on this.
The commutativity hypothesis is unnecessary. Let $A\backslash G/B=\{AgB:g\in G\}$ be the space of double cosets. There is a map $G/A\times G/B\to A\backslash G/B$ given by $(aA,bB)\mapsto Aa^{-1}bB$. The fibers of this map are precisely the $G$-orbits of $G/A\times G/B$. Letting $T$ be any system of $A$-$B$ double coset representatives, the elements $(t^{-1}A,B)$ or else the elements $(A,tB)$ form a corresponding system of representatives for the $G$-orbits in $G/A\times G/B$. There is no canonical set of reps for the double coset space any more than there is for a single-side coset space $G/H$.