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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.

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Represent by permutation matrices and use Maschke's theorem? –  Mark Sapir Oct 4 '10 at 11:55

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up vote 4 down vote accepted

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 $G_{0}$.

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Yes, this is pretty much what I was looking for. Thanks! –  Colin Reid Oct 4 '10 at 14:36

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