Can the symmetric groups on sets of different cardinalities be isomorphic?

Question

For any set X, let S_X 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 S_X is isomorphic to S_Y?

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 S_X 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 2^κ < 2^λ for all cardinals, (in another words, if the exponential function is one-to-one, a weakening of the GCH), then again S_κ is not isomorphic to S_λ 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 2^κ = 2^λ for some cardinals κ < λ. In this case, we will have two different cardinals κ < λ, whose corresponding symmetric groups S_κ and S_λ 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 2^ω = 2^ω₁. But also, if one just forces ¬CH by adding Cohen reals over a model of GCH, then again 2^ω = 2^ω₁.

(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.)

Answer 1

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 $X$.

Answer 2

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.

Answer 3

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.

Answer 4

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

Steve

Answer 5

I figured out a solution that just takes some basic combinatorics, and doesn't use the axiom of choice at all; I'm surprised no one else posted something similar already.

Assume $X$ is not finite, since we can handle that case easily.

The transpositions in $S_X$ can be distinguished as follows: They are the conjugacy class $T$ consisting of elements of order 2 with the property that the set of products of pairs of elements of $T$ contains no more than one conjugacy class of order 2. To see that no other conjugacy class satisfies this property, observe that for any product $s$ of at least two disjoint transpositions (including possibly infinitely many), one can select a subset of two of them and interchange two of the letters to obtain $s'$ such that $ss'$ is a product of exactly two disjoint transpositions (using the Klein 4 group in $S_4$). If $s$ is a product of exactly 2 or 3 transpositions then we can multiply by a disjoint product of 2 or 3 transpositions to obtain another conjugacy class. If s is a product of 4 or more transpositions, then we can select another two transpositions in s to obtain $s''$ such that $ss''$ is a product of exactly four disjoint transpositions.

Next we aim to characterize letters as equivalence classes of pairs of elements of $T$ which multiply to an element of order 3, so that the pair $\{(a, b), (b, c)\}$ distinguishes $b$. We need to describe the equivalence relation on these pairs. We can express the transposition $(a, c)$ as a conjugate of one element of the pair by the other. Therefore when we have another pair $\{(a', b'), (b', c')\}$, we can express the condition that $\{a, c\}$ is disjoint from $\{a', c'\}$. In this case we say that the two pairs are related if multiplying any two of the four transpositions gives an element of order 3. Then we take the transitive closure of that relation.

So we have constructed a set from the abstract group $S_X$ which is in natural bijection with $X$. Note that the argument distinguishing $T$ works when $|X| \ge 12$, and the argument characterizing the equivalence relation works when $|X| \ge 7$. There's probably a clever way to adapt both arguments for fewer elements; this is just what I came up with. Note that the transpositions cannot be distinguished in $S_6$.

Answer 6

Maybe too late as for a question asked 11 years ago, but I would like to inform that this paper contains a topological proof of the following theorem which is related to the answer of Mariano Suárez-Álvarez.

Theorem. For any cardinals $\kappa<\lambda$, the group $Alt(\lambda)$ of even finitely supported permutations of $\lambda$ is not isomorphic to a subgroup of the permutation group $Sym(\kappa)$.