# Thread: Equivalence relation and order of each equivalence class

1. ## Equivalence relation and order of each equivalence class

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

2. The "order of the equivalence classes" is just the number of elements in each equivalence class. And it is easy to prove that all equivalence classes, for a given equivalence relation, have the same number of elements so your answer will be a single number. Since you are dealing with groups, it might be simplest to find how many pairs (r, k) are equivalent to $(e_H, e_K)$, the identity elements of each group.