Relationships between Symmetric Groups Here is a question that I can't find the answer to in my notes or my textbook.
Is there any relation between $\text{Sym}_n$ and $\text{Sym}_k$ where $k < n$? Is $\text{Sym}_k \subset \text{Sym}_n$? Since $k! |  n!\text{ } \forall  k < n $ they satisfy Lagrange's theorem, but of course that doesnt guarantee the existence of a subgroup
I'm trying to use this result to show that that there is a transitive action of a finite group $G$ on a smaller finite set $S$ by showing that $G$ or a subgroup of $G$ is isomorphic to $\Sigma (S)$ (The permutation group of $S$) which has a natural transitive action. 

Yes. $Sym_k < Sym_n$ for $n < k$ and it corresponds to a subgroup of all permutations with the same $k-n$ elements.
(That is, fix $k-n$ elements first, then look at all the transpositions that keep them in place. This subgroup of $S_k$ is isomorphic to $S_n$.)
