21.5 (2026)

Let $p$ be a prime. Let $G$ be a transitive subgroup of the group of finitary permutations $\text{FSym}(\Omega)$ of a set $\Omega$, let $N$ be a normal subgroup of $G$, and let $S$ be a transitive Sylow $p$-subgroup of $G$.
$\qquad$ (a) Is it true that $S \cap N$ is a Sylow $p$-subgroup of $N$?
$\qquad$ (b) Is it true that $SN/N$ is a Sylow $p$-subgroup of $G/N$?
$\qquad$ (c) Are any two transitive Sylow $p$-subgroups of $G$ locally conjugate in $G$?

Two subgroups $X, Y$ of a group $G$ are said to be locally conjugate if there is a locally inner automorphism $\varphi$ of $G$ such that $X^\varphi = Y$. An automorphism $\varphi$ of $G$ is said to be locally inner if for every finite subset $A \subseteq G$ there is an element $g = g(A) \in G$ such that $a^\varphi = g^{-1} a g$ for all $a \in A$.