For $X$ a finite set one can show easily that the abelization of the symmetric group on $X$ is given by the group of order $2$ and that the commutator subgroup of $\operatorname{Sym}X$ is given by the group of even permutations:
Proof: Consider the $\operatorname{sign}\colon\operatorname{Sym}X\longrightarrow\{\pm1\}$. Since the two-element-group is commutative, we find $[\operatorname{Sym}X,\operatorname{Sym}X]\subseteq\ker\operatorname{sign}=\operatorname{Alt}X$. Since the symmetric group is not commutative, the commutator subgroup is a nontrivial normal subgroup of the alternating group and by simplicity of the latter, we find that the commutator subgroup is equal to $\operatorname{Alt}X$. (The cases $\operatorname{card}X=1,2,4$ of course need a slight variation because of lacking noncommutativity or simplicity of the alternating group).
Question 1: For $X$ infinite, I have read that the abelization of $\operatorname{Sym}X$ is trivial. Is there an elementary way to see this? Elementary in particular means that the argument makes no use of the classification of normal subgroups of the symmetric group.
Question 2: Let $I$ be the category asssociated to the preordered set of finite cardinals and denote by $F\colon I\longrightarrow\mathbf{Grp}$ the functor which associates to each cardinal $\alpha$ the symmetric group on $\alpha$. The colimit of this functor is then isomorphic to the group of permutations with support stricitly less then $\aleph_0$, which is usually called the finitary symmetric group and which I denote by $\operatorname{Sym}_{\aleph_0}X$ for $X$ a set of cardinality $\ge\aleph_0$. Since the abelization functor is left adjoint to the natural fucntor $\mathbf{Ab}\longrightarrow\mathbf{Grp}$ and thus commutes with colimits, the abelization of the finitary symmetric group is given by the group of order $2$, aswell. My second question is, wether this argument can be generalized to arbitrary cardinal numbers.
Maybe my questions are simple to answer, but since I don't know a lot of either category theory, or set theory or algebra, I need some help.