Proving |HK| = |H||K|/|H∩K| in Group Theory

[rakalakalili]

Homework Statement

If G is a group, and H and K are subgroups of G, prove:
$$|HK|=\frac{|H||K|}{|H\cap K|}$$

Homework Equations

The Attempt at a Solution

A basic counting argument can be made by saying that HK consists of an element of H with an element of K, so you have |H||K| elements. What I don't quite see is how this creates over counting by a factor of |H n k|. A hint would be much appreciated!

 

[Office_Shredder]

If h is in H and k is in K,

hk=hg g^-1k

For any choice of g. In particular, if hg is in H and g^-1k is in K, then we've counted hk twice.

 

[rakalakalili]

So if hg is in H, then that implies g is in H, which implies that g-1 is also in H. A similar argument shows that g and g-1 are in K. So for each element g of HK, hk will be equal to hgg^-1k, and that is where the extra count come in. Thank you!