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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 fi nite 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.

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  • $\begingroup$ The second question is covered at math.stackexchange.com/questions/168942/… $\endgroup$ – Arnaud D. Nov 7 '18 at 10:41
  • $\begingroup$ Oh ... Many thanks $\endgroup$ – Nawal Nov 7 '18 at 10:43
  • $\begingroup$ As for the first one, to find the order of the stabilizer you can try to prove that it is isomorphic to $S_k\times S_{n-k}$. $\endgroup$ – Arnaud D. Nov 7 '18 at 10:48

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