# Groups

• March 23rd 2009, 06:44 AM
math_help
Groups
http://i453.photobucket.com/albums/q...188/math19.jpg

Stuck on this question, struggling to see where to start.
• March 23rd 2009, 11:16 AM
Opalg
Each element of HK is of the form hk, where $h\in H,\;k\in K$. But there may be more than one way of representing the same element as such a product.

So suppose that $h_1k_1 = h_2k_2$. Then $h_2^{-1}h_1 = k_2k_1^{-1} = p$ say, where $p\in H\cap K$ (because $p = h_2^{-1}h_1\in H$ and also $p = k_2k_1^{-1}\in K$).

Use that to count the number of ways that each element of HK can be expressed as a product of the form hk.
• March 24th 2009, 02:55 AM
Halmos Rules
• March 25th 2009, 11:39 AM
math_help
You could take $G = S_3$, with H and K the subgroups of order 2 generated by the transpositions (12) and (13) respectively.