Problem to solve:

Let X and Y be nonempty subsets of a group G, and XY denotes the set {g $\displaystyle \in$ G| g = xy for some x $\displaystyle \in$ X, y $\displaystyle \in$ Y}.

Let H and K be subgroups of a finite group G and define a relation on the set H $\displaystyle \times$ K by setting (r,s) ~ (t,u) if rs = tu $\displaystyle \forall$ r, t $\displaystyle \in$ H, s,u $\displaystyle \in$ K.

Part 1: Show that this is an equivalence relation such that each equivalence class has order |H $\displaystyle \cap$ K|.

Part 2: Prove |HK|$\displaystyle \cdot$ |H $\displaystyle \cap$ K| = |H| $\displaystyle \cdot$ |K|

Hello,

I need help with the problems above. I'm also a little confused on part 1...I know how to show the equivalence relation for the set H $\displaystyle \times$ K, but then the problem stated something about the order of equivalence class. How am I going to show that? Appreciate anyone's help on this. Thank you in advance