I have two similar questions:
Justify that a set of n elements has $\binom nk$ subsets of k elements by using the orbit-stabilizer theorem.
The orbit-stabilizer theorem says that $|Orb(x)|=[G:Stab(x)]=|G||Stab(x)|$
My attempt is if $X = \{1,2,3,...,n \} $ and S be the set of all subsets of X with k elements, then $S_{n}$ acts on $S$ with $n!$ permutations.
The orbit of $\{1,2,3, ...,k \}$ is S.
The order of the stabilizer should be $k!(n-k)!$(I do not know how to prove that!).
The second question:
For finite subgroups $H$ and $K$ of a group G, show that : $|HK| = \frac{|H||K|}{|H\cap K|}$ using the orbit-stabilizer theorem.
My attempt: with the direct product $H \times K$, we have $(h,k)*g := hgk^{-1}$.
Any idea on whether it is right or wrong.
