My problem is in proving the following:
The text suggests that I construct the functionLet H be a subgroup of a group G. Then |Hg| = |H| = |gH| for all g in G.
for all x where and . All the text says is that f is "easily shown to be a bijection."
First, this looks suspiciously like a projection map of some kind. I'll tackle that later...I should be able to show that f is a surjection.
The problem I am having is this: f maps a set of cosets onto a single element of H. Okay, but looking at a couple of examples of the function acting explicitly on a given group:
and H < G then
This implies that f is multiple valued, ie f(a) is equal to both e and a. Which can't be right.
Should I be taking the function of a coset, , in which case hx is a representative of that coset? I'm obviously looking at something wrong, but I don't know what.