A question on subgroups of an infinite permutation group

A question on subgroups of an infinite permutation group

I have come across this problem in the course of studying for my
comprehensive exam in algebra, which is fast approaching. I would
appreciate any suggestions and/or hints as to its solution. Let $S$ be an
infinite set, and let $G:=\mathrm{Sym}(S)$, the symmetric group on $S$.
Consider the subgroup $A<G$ given by
$A:=\left\langle\left(a\hspace{2pt}b\right)\left(c\hspace{2pt}d\right)\Big|a,b,c,d\in
S\right\rangle$, where the notation $\left(x\hspace{2pt}y\right)$ denotes
the transposition of the elements $x$ and $y$. Prove that $A$ is a simple
group. The subgroup $A$ is, essentially, the analogue of the alternating
group for finite sets $S$. However, I have as yet been unable to figure
out how to apply the proofs of the simplicity of $A_n$ for $n\geq 5$ to
this problem (if that is even a valid strategy). Any suggestions would be
appreciated.