On Sylow subgroups of finitary symmetric groups $\DeclareMathOperator\FSym{FSym}$Let $p$ be a prime and $S$ be a transitive Sylow $p$-subgroup of $\FSym(\mathbb{N})$, the finitary symmetric group of the set of all natural numbers.
Question: Is $S$ totally imprimitive and uniserial (i.e. for every $i\in \mathbb{N}$, there exists a unique system of imprimitivity on $\mathbb{N}$ with blocks of size $p^i$—this concept is introduced by P.M. Neumann in [The structure of finitary permutation groups, Arch. Math. 27 (1976), 3-7 DOI link])?
 A: Yes: all transitive Sylow $p$-subgroups of $\mathrm{FSym}(\mathbb{N})$ are permutation isomorphic to the infinite wreath product $C_p \wr C_p \wr \ldots $. This permutation group is uniserial and has a unique system of blocks of size $p^i$ for each $i \in \mathbb{N}$; one such block is an orbit of the canonical subgroup isomorphic to $C_p \wr \ldots \wr C_p$ (with $i$ factors) used to construct the wreath product.
For a reference, see (2) on page 422 of this paper by Agnieszka Bier, Yuriy Leshchenko and Vitaliy Sushchanskyy.
(Just to be clear, here $C_p \wr C_p \wr \ldots $ is the restricted wreath product, with the left-most $C_p$ corresponding to blocks of size $p$, not the profinite wreath product $\ldots \wr C_p \wr C_p$. The latter is the automorphism group of the infinite $p$-ary rooted tree in which the right-most $C_p$ corresponds to the action on the $p$ branches below the root. It does not contain any finitary permutation except the identity.)
Edit. Since the linked paper proves far more than we need (it's really about automorphism groups of directed limits of trees), let me add a proof using only Wielandt's theorem that a finitary primitive permutation group on an infinite set $\Omega$ is either $\mathrm{FSym}(\Omega)$ or its index $2$ subgroup $\mathrm{Alt}(\Omega)$. Since neither is a $p$-group, it follows that a maximal $p$-subgroup $P$ of $\mathrm{FSym}(\mathbb{N})$ is imprimitive.
It is well known that every block of a imprimitive finitary permutation group is finite. (Proof. If $\Gamma$ is a block then $\Gamma g \cap \Gamma = \varnothing$ for any $g$ moving $\omega \in \Gamma$ to $\omega g \not\in \Gamma$, so $\Gamma \subseteq \mathrm{supp}\ g$.) Moreover, if $P$ has a maximal proper block $\Delta$ then $P$ acts primitively on the set $\{\Delta g : g \in P\}$, again contradicting that $P$ is a $p$-group. Therefore there is an infinite chain of finite blocks $\Gamma_1 \subset \Gamma_2 \subset \ldots $ and $P$ embeds in the restricted iterated wreath product $S_{\Gamma_1} \wr S_{\Gamma_2} \wr \ldots$ where, since $P$ is a $p$-group, each $S_{\Gamma_i}$ is a finite $p$-group. By maximality, each $S_{\Gamma_i}$ is a Sylow $p$-subgroup of a finite symmetric group, so of the expected form $C_p \wr \ldots \wr C_p$. In particular $P$ is totally imprimitive and uniserial.
