Let $G$ be a transitive permutation group on a set of size $n$, and suppose $Z(G)=1$ (for instance $G$ is a direct power of a non-abelian simple group). What can we say about the centraliser $K$ of $G$ in $Sym(n)$? I'm interested firstly if there are any restrictions on $K$ independent of degree, and secondly on what role the degree plays.



Centralisers of transitive permutation groups
Of course, there is the classical result that $C\_{Sym(n)}(G)$ is a semi-regular subgroup of $Sym(n)$ of cardinality $|Fix(G\_{0})|$, where $G\_{0}$ is the stabilizer of a point and $Fix(G\_{0})$ is the set of points fixed by $G\_{0}$.


