Subgroups of the symmetry group

Question

Question:

Let $A = \{\{x,y\}:x,y\in\{1,2,...,n\},x\neq y\}$. We want to show that $\exists G\leq \text{Sym(A)}$ s.t. $G\cong S_n$ and $\{g(\{1,2\}):g\in G\}=A$

Attempt:

For the first part I'm pretty sure I have to use the first isomorphism theorem which will give that for a homomorphism $f:C\rightarrow D$, $C/\text{Ker}f\cong \text{Im}f\leq D$. In this case I think that D will be Sym(S) however I'm completely lost as to what the group C would be and what exactly the homomorphism would be. From there I'm assuming the image will likely be something that is clearly isomorphic to $S_n$. For the second part I'm thinking that the homomorphism should make it clear what the Coset is.

Answer 1

Let $S_n$ act on $A$ in the following way:

$\sigma\cdot\{x,y\}=\{\sigma(x),\sigma(y)\}$.

This group action is transitive and faithful.

Do you know how group actions on $X$ relate to $\text{Sym}(X)$?