$\DeclareMathOperator\FSym{FSym}$Let $\FSym(\mathbb{N})$ denote the finitary symmetric group on the set of natural numbers. How many Sylow $p$-subgroups does $\FSym(\mathbb{N})$ have for any prime $p$? Countably or uncountably many?



Sylow $p$-subgroups of FSym($\mathbb N$)
Uncountably (continuum) many. A $p$-Sylow subgroup (or at least some of them) determines a nested partition (into $p$-element subsets, into $p^2$-element subsets, etc), and hence determines a partition into $p$-element subsets. There are continuum many such partitions and all are conjugate under the permutation group.


