Maximal subgroups of $S_{\Bbb N}$ and $A_{\Bbb N}$

Question

Let $S_{\Bbb N}$ be the countable symmetric group of all permutations of the naturals with finite support and $A_{\Bbb N}$ --- the corresponding alternating group. How to describe all maximal subgroups of these groups which are isomorphic to $S_{\Bbb N}$? Is it possible to obtain the answer from well-known answers on the same question for the finite case?

Answer 1

Here I will follow the notation from Dixon--Mortimer, rather than OP's, because it seems to be more standard, and clearly distinguishes the full and finitary symmetric groups.

Let $\Omega$ be an infinite set. By Dixon--Mortimer (1996), Lemma 8.3A, the only primitive subgroups of $\operatorname{FSym}(\Omega)$ are $\operatorname{FSym}(\Omega)$ and $\operatorname{Alt}(\Omega)$ (briefly, by a classical result of Jordan, if a primitive group $G$ contains an element of support $m$ then $|\Omega| \le (m-1)^{2m}$ unless $G \ge \operatorname{Alt}(\Omega)$). Moreover, for any imprimitive (transitive) subgroup of $\operatorname{FSym}(\Omega)$, the blocks must be finite. Therefore the maximal subgroups of $\mathrm{FSym}(\Omega)$ are:

• maximal intransitive groups $\operatorname{FSym}(X) \times \operatorname{FSym}(Y)$, where $\Omega = X \sqcup Y$ is a nontrivial partition of $\Omega$ into proper nonempty subsets $X$ and $Y$,
• maximal imprimitive groups $\bigoplus_{i \in I} \operatorname{Sym}(\Delta_i) \rtimes \operatorname{FSym}(I)$, where $\Omega = \sqcup_{i \in I} \Delta_i$ is an equipartition of $\Omega$ into a collection of finite nonempty subsets $\Delta_i$ of equal cardinality,
• $\operatorname{Alt}(\Omega)$.

The only groups on this list $\cong \operatorname{FSym}(\Omega)$ are the point stabilizers $\operatorname{FSym}(\Omega)_x$.

The same list holds for maximal subgroups of $\operatorname{Alt}(\Omega)$, taking the intersection with the alternating group. The only further groups we get $\cong \operatorname{FSym}(\Omega)$ are stabilizers of a pair of points $\operatorname{Alt}(\Omega)_{\{x,y\}}$.

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Old answer, really more about the full symmetric group rather than the finitary one:

Some information is in Dixon--Mortimer (1996), Chapter 8.5, "Maximal subgroups of the symmetric groups". Unfortunately:

The situation for infinite symmetric groups is more complicated, and it seems unlikely that there is a satisfactory description of the maximal subgroups in this case.

I do not think the situation has changed dramatically since 1996.

These lecture notes of Peter Neumann are also worth looking at: https://arxiv.org/abs/2307.11564