On non-trivial normal subgroup(s) of $A(S)$ , where $S$ is infinite

Question

Let $A(\mathbb R)$ be the permutation group of $\mathbb R$ , is this group simple ? In general for an infinite set $S$ , how may we determine whether $A(S)$ has any non-trivial normal subgroup or not ?

Answer 1

If $S$ is an infinite set, the group of finitary permutations on $S$, that is, those permutations that fix all but a finite number of elements, is always a non-trivial, proper normal subgroup.

See this page from Peter Cameron's Permutation Groups for further details. It mentions in particular the finitary alternating groups. And the bounded symmetric groups, that is, fix a cardinal number $\mathfrak{n}$ not larger than the cardinality of $S$, and consider the group of permutations that move less than $\mathfrak{n}$ points.