Normal subgroups of infinite symmetric group
For a general infinite set $X$, the normal subgroups of Sym$(X)$ are:
Sym$(X)$;
the trivial subgroup;
the even permutations of $X$ with finite support;
for each cardinality $c$ with $\aleph_0 \le c \le |X|$, the group of all permutations of $X$ with support less than $c$.
There is a straightforward proof in Chapter 8 of the book "Permutation Groups" by J.D. Dixon and B.M. Mortimer, where the result is attributed to Baer.
I don't think the proof uses CH or GCH although the result itself is affected by CH.
You are inquiring about the Schreier-Ulam theorem. This old MO post contains an answer of mine with the statement of the result; here is a link to the original paper (thanks to t.b.). I would be happy to supplement this and/or that answer with a link to a free, electronically available English language proof, if anyone knows one.
The widely known reference for this result is Schreier-Ulam (1933). But this was previously proved by Luigi Onofri (1929), in the third of a series of 3 articles, of 1927, 1928, and 1929, not mentioned in Schreier-Ulam's article, in which he also introduces (§4 of the first opus) its natural topology on the group of permutations of a countable set.
L. Onofri. Teoria delle sostituzioni che operano su una infinità numerabile di elementi. Memoria 3a. Annali di Matematica Pura ed Applicata December 1929, Volume 7, Issue 1, pp 103–130. (restricted access: Springerlink).
This is in §141, p124:
I gruppi $G_1$ e $G_2$ sono gli unici sottogruppi invarianti del totale $G$. (The groups $G_1$ and $G_2$ are the unique invariant subgroups of the whole group $G$.)
Here $G$ is the group of permutations of an infinite countable set $I$, $G_1$ its subgroup of finitely supported permutations ("substitutions operating on finitely many elements" in his language) and $G_2$ the subgroup of even elements in $G_1$, and "invariant" means invariant under conjugation.
Onofri's proof has 3 steps (I try to use italics when I translate in English without translating in modern math language):
(a) he defines (§100, p104) a subset $H$ of $G$ to be infinitely transitive of infinite grade if it is transitive by pre-composition of injective self-maps whose image with infinite complement, or, to be closer to his language, if for any two infinite systems of (distinct) elements $[x_1,x_2,\dots]$, $[y_1,y_2,\dots]$ in $I$ such that the complements (called residual systems) of both $\{x_i:i\ge 1\}$ and $\{y_i:i\ge 1\}$ are infinite, there exists $s\in H$ ("a substitution $s$ of $H$") such that $s(x_i)=y_i$, $(i=1,2,\dots)$. Here "infinite grade" refers to infiniteness of the complement.
He proves (§110, p108) that the only subgroup of the group of permutations of $I$ that is infinitely transitive of infinite grade, is the whole group of permutations.
(b) He proves (§136, p121) that the only normal subgroup not contained in the subgroup of finitely supported permutations, is the whole group, and his strategy consists in proving that such a subgroup is infinitely transitive of infinite grade.
His statement is actually Un complesso C contenente una sostituzione h su infiniti elementi e le sue trasformate mediante G, coincide con il totale.
A literal translation is: A complex containing a substitution h on infinitely many elements and its conjugate transforms G coincides with the total.
An adapted translation, from my understanding, is: "A subsemigroup (of $G$, the group of permutations of $I$) containing an infinitely supported permutation as well as its conjugates coincides with the whole group of permutations."
Here I should explain why I'm translating complesso (literally "complex") into "subsemigroup": "complesso" is evoked, not defined in the first memoir (1927, §25, p89): indeed he defines a somewhat inefficient terminology for subsemigroups of the group of permutations "(sub)groups" for subgroups, "pseudogroups" those subsemigroups that are not subsemigroups, and even the splits pseudogroups as "simple" and "composite", where "simple" means that no element is invertible. Then he seems to use "complex" means something which either a group, a "simple" pseudogroup, or a "composite pseudogroup". I'm still a bit confused since he occasionally uses "complesso" for cosets, but anyway the proof is convincing if one interprets as "subsemigroup".
(c) The hardest part is done, since the result now reduces to classify the normal subgroups contained in the subgroup of finitely supported permutations; this short concluding step is §141, p124. Actually, he then (§142) observes that the alternating subgroup is the "only invariant subgroup" of the group of finitely supported permutations while the alternating subgroup itself has "no normal subgroup except the identity". It's a bit sloppy on whether he considers the trivial/whole subgroups as normal ("invariant") subgroups, but the proof is totally correct as far as I could check.
He, by the way, observes (§139) after step (b) the corollary that the group of permutations of $I$ has no proper subgroup of finite index, and in particular (§140) that there is no way to coherently extend parity of finite permutations to arbitrary permutations (since this would produce a subgroup of index 2).