Normal subgroups of infinite symmetric group

I recently took a course on group theory, which mentioned that the following proposition is equivalent to the continuum hypothesis: "The infinite symmetric group (i.e. the group of permutations on the set $\mathbb{N}$) has exactly 4 normal subgroups." Does anyone have any references or explanation for this?

For a general infinite set $X$, the normal subgroups of Sym$(X)$ are:

1. Sym$(X)$;

2. the trivial subgroup;

3. the even permutations of $X$ with finite support;

4. 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.