$\DeclareMathOperator\FSym{FSym}$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](https://doi.org/10.1007/BF01224634)])?



On Sylow subgroups of finitary symmetric groups
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](https://www.sciencedirect.com/science/article/pii/S002186931600017X) 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.


