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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?

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Well, one of the normal subgroups is the set of permutations of finite support. –  Arturo Magidin Jul 4 '12 at 0:58
    
The support are the elements that are not fixed by the permutation. I.e., I'm refering to the subgroup of permutations that are the identity on a cofinite subset of $\mathbb{N}$. –  Arturo Magidin Jul 4 '12 at 1:01

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up vote 7 down vote accepted

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.

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How does this relate to CH? Does this theorem assume it? As I see it, it seems that the theorem, if true with no assumptions, would imply that the statement in question is false... –  tomasz Jul 4 '12 at 1:18
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@tomasz: CH does not enter into it. Schreier-Ulam states that the only nontrivial normal subgroups of $S_{\Omega}$ are $\cup A_n$ and $\cup S_n$. The other two (to make up the count of four) are the trivial and total subgroup. –  Arturo Magidin Jul 4 '12 at 1:27
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The original paper by Schreier and Ulam is available here for those who read German. The homepage of the Polish Virtual Library is certainly worth bookmarking. –  t.b. Jul 4 '12 at 1:28
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@mlbaker: Seems like it. On the other hand, I'd be surprised if counting the normal subgroups of permutations of $\mathbb{R}$ was not contingent on CH. One would expect (at least) a normal subgroup of the form "all permutations fixing $\aleph_\alpha$ elements" for each $\aleph_\alpha < \mathfrak{c}$. –  Jason DeVito Jul 4 '12 at 2:14
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@mlbaker: I don't see how CH would come into it in any case. The arguments are all finitistic, since we are dealing with groups. –  Arturo Magidin Jul 4 '12 at 4:07

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.

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I don't have Dixon and Mortimer available, but I think the original reference is R. Baer, Die Kompositionsreihe der Gruppe aller eineindeutigen Abbildungen einer unendlichen Menge auf sich, Studia Math. 5 (1934), 15-17. –  t.b. Jul 4 '12 at 11:35

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