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For any set X, let SX be the symmetric group on X, the group of permutations of X.

My question is: Can there be two nonempty sets X and Y with different cardinalities, but for which SX is isomorphic to SY?

Certainly there are no finite examples, since the symmetric group on n elements has n! many elements, so the finite symmetric groups are distinguished by their size.

But one cannot make such an easy argument in the infinite case, since the size of SX is 2|X|, and the exponential function in cardinal arithmetic is not necessarily one-to-one.

Nevertheless, in some set-theoretic contexts, we can still make the easy argument. For example, if the Generalized Continuum Hypothesis holds, then the answer to the question is No, for the same reason as in the finite case, since the infinite symmetric groups will be characterized by their size. More generally, if κ < λ implies 2κ < 2λ for all cardinals, (in another words, if the exponential function is one-to-one, a weakening of the GCH), then again Sκ is not isomorphic to Sλ since they have different cardinalities. Thus, a negative answer to the question is consistent with ZFC.

But it is known to be consistent with ZFC that 2κ = 2λ for some cardinals κ < λ. In this case, we will have two different cardinals κ < λ, whose corresponding symmetric groups Sκ and Sλ nevertheless have the same cardinality. But can we still distinguish these groups as groups in some other (presumably more group-theoretic) manner?

The smallest instance of this phenomenon occurs under Martin's Axiom plus ¬CH, which implies 2ω = 2ω1. But also, if one just forces ¬CH by adding Cohen reals over a model of GCH, then again 2ω = 2ω1.

(I am primarily interested in what happens with AC. But if there is a curious or weird counterexample involving ¬AC, that could also be interesting.)

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The normal subgroups of $S(X)$ are the subgroup of elements which move finitely many points and its subgroup of even permutations, and the subgroups $S_\kappa(X)$ of permutations which move at most $\kappa$ points; this is the Schreier–Ulam–Baer theorem. By looking at the order type of the lattice of normal subgroups (which is a chain) maybe you can guess the cardinal of $X$. –  Mariano Suárez-Alvarez Jan 25 '10 at 16:14
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there are no finite examples ... um, 0! = 1!, right? –  Gerald Edgar Jan 25 '10 at 18:07
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@Gerald: Yes, I should ask about nonempty sets X, Y. –  Joel David Hamkins Jan 25 '10 at 18:30
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@Michael: What could be more natural than mapping the identity of one group to the identity of another, especially when the groups have no other elements? –  Joel David Hamkins Jan 25 '10 at 19:40
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@Scott: What canonical map of permuation groups are you thinking about? I don't know any and a have no idea how to turn a general map $X\to Y$ into a map $S(X)\to S(Y)$. Injective maps work, okay. But what is with the general case? –  Johannes Hahn Mar 18 '10 at 13:22

4 Answers 4

up vote 30 down vote accepted

According to the Schreier–Ulam–Baer theorem, the nontrivial normal subgroups of $S(X)$ are (i) the subgroup $S_\mathrm{fin}(X)$ of permutations of $X$ of finite support, (ii) the subgroup $A_\mathrm{fin}(X)$ of $S_\mathrm{fin}(X)$ of even permutations, and (iii) for each cardinal $\kappa$ the subgroups $S_{<\beta}(X)$ and $S_{\leq\beta}(X)$ of permutations which move strictly less than $\beta$ points and at most $\beta$ points, respectively.

Since, as you said, a cardinal is determined by the order type of the set of cardinals below it, looking at the lattice of normal subgroups of $S(X)$, then, lets you guess the cardinal of $X$.

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Thanks for the great answer! But it seems a funny way to me to make the list, for it is redundant in several ways. First, item (i) is included in the first part of item (iii) in the case beta=omega as S_{<omega}. Also, the second part of item (iii) is a special case of the first part, since S_{\leq\beta}(X)=S_{<\beta+}(X). So I think you can omit item (i) and the second part of (iii) completely, and have the same subgroups. (Also, you have a typo, since \kappa should be \beta.) –  Joel David Hamkins Jan 25 '10 at 23:30

In an ancient paper with Saharon Shelah, I proved that if κ < λ, then Sym(λ) does not embed into Sym(κ). The proof is based on results in an even more ancient paper with John Dixon and Peter Neumann. The relevant papers are:

Saharon Shelah and Simon Thomas, Implausible subgroups of infinite symmetric groups. Bull. London Math. Soc. 20 (1988), no. 4, 313--318.

John D. Dixon, Peter M. Neumann and Simon Thomas, Subgroups of small index in infinite symmetric groups. Bull. London Math. Soc. 18 (1986), no. 6, 580--586.

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Simon, Welcome to Math Overflow! I believe that you have a lot to contribute here, and the community will learn a lot from you. –  Joel David Hamkins Mar 18 '10 at 1:27

The following seems simpler than the answers given earlier. I apologize if this answer (or a simpler one) is already in one of the links and I overlooked it.

I claim that, for any infinite set $X$, the cardinality $|X|$ can be obtained from the symmetric group Sym$(X)$ as the smallest cardinality of any conjugacy class other than the trivial class $\{1\}$. First, there is a conjugacy class of size $|X|$, for example, the class of those permutations that just interchange two elements of $X$ while fixing everything else.

Now the main point: Suppose $C$ is a non-trivial conjugacy class. Consider any element $\pi$ of $C$ and any element $x\in X$ moved by $\pi$. (Such an $x$ exists as $C$ isn't the trivial class.) For any element $y\in X-\{x\}$, consider the permutation that sends $\pi(x)$ and $y$ to each other and fixes everything else (including, in particular, $x$). Then $\sigma\pi\sigma^{-1}$ (also known as $\sigma\pi\sigma$) is a conjugate of $\pi$ that sends $x$ to $y$. So the $|X|$ different possible $y$'s give us $|X|$ different conjugates of $\pi$. Therefore, $|C|\geq|X|$, as claimed.

Remark: I used the Axiom of Choice twice here, first to say that the number of pairs from $X$ is $|X|$, and second to say that the number of possible $y$'s is also $|X|$. (The second fact follows easily from the first.) I don't know whether the result holds in the absence of the Axiom of Choice.

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This is a very nice argument. I have a feeling that what it gives (with a little extra massaging) is not so far from the Schreier-Ulam-Baer theorem... –  Igor Rivin Apr 23 '12 at 14:16

See exercises 4.6.5 - 4.6.8 in Dummit & Foote, 3rd edition. In particular, the Schreier-Ulam theorem is not needed.

Steve

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+1: Wow, the newest edition of D&F is impressively comprehensive. Their proof goes by establishing part of B-S-U: namely that the index 2 alternating subgroup A of the subgroup of all permutations moving only finitely many elements is the unique nontrivial minimal normal subgroup of S(X). Since A and X have the same cardinality, this suffices. –  Pete L. Clark Jan 30 '10 at 17:16
    
Thanks, Pete! And also Steve. –  Joel David Hamkins Jan 31 '10 at 2:25

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