# Equivalence relation and order of each equivalence class

• Sep 29th 2009, 04:47 PM
anlys
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 $\in$ G| g = xy for some x $\in$ X, y $\in$ Y}.
Let H and K be subgroups of a finite group G and define a relation on the set H $\times$ K by setting (r,s) ~ (t,u) if rs = tu $\forall$ r, t $\in$ H, s,u $\in$ K.
Part 1: Show that this is an equivalence relation such that each equivalence class has order |H $\cap$ K|.
Part 2: Prove |HK| $\cdot$ |H $\cap$ K| = |H| $\cdot$ |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 $\times$ 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 :)
• Sep 30th 2009, 10:03 AM
HallsofIvy
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.