I want to prove that |HK| = |H||K| / |H ∩ K| using group actions. First, I let H and K be subgroups of G and the set S be defined as the set of left cosets of K in G. I define a group action by a * (xK) = axK where and .
by considering the orbit of the element K in S, i have that the orbit of K is the set of all s = a*K = aK, such that . but this is just the set theoretic product HK, so the orbit of K is equal to HK. From the orbit stabilizer theorem, the number of elements in the orbit of K is equal to the number of left cosets of the stabilizer of K. The stabilizer of K is the set of all a such that a*K = aK = K and . but this implies that and so . so the stabilizer of K is the set .
now the number of left cosets of the stabilizer of K is |H|/|H ∩ K|. therefore the orbit stabilizer theorem says that |HK| = |H|/|H ∩ K|, but i seem to be missing a factor of |K| somewhere and going through the problem I can't seem to find where it must come from. Can someone help show me what went wrong? thanks.