21.6 (2026)

Let $p$ be a prime. A totally imprimitive $p$-group $H$ of finitary permutations is said to have the cyclic-block property if in the cycle decomposition of every element the support of every cycle is a block for $H$. Let $G$ be a transitive subgroup of the group of finitary permutations $\text{FSym}(\Omega)$ of a set $\Omega$. Does every transitive Sylow $p$-subgroup of $G$ contain a transitive subgroup which has the cyclic-block property?