Automorphism group of the Alternating Group - a proof
In the above question, Derek Holt asserts that a primitive permutation groups has trivial centraliser in the symmetric group. Since I could not understand why, I thought to open a new question (maybe someone like me could be interested in this). Is there any simple proof of this?
I'm not sure whether clarifying an answer to an earlier question merits a formal answer ( and I have incorporated a minor point made by Derek in this answer anyway), but here goes : First of all, groups of prime order $p$ in $S_{p}$ are an exception, as Derek pointed out in his answer to the previous question. Suppose then that $G$ is a primitive subgroup of $S_n$ which is not of prime order, and let $X$ be the centralizer of $G.$ Then $G$ permutes the orbits of $X$, so if $X≠1$, then $X$ is transitive. Now $X$ permutes the fixed points of $g$ for any $g∈G$, so no non-identity element of $G$ fixes any point. Hence $G$ is regular (since it is transitive). Hence the trivial subgroup of $G$ is maximal, so $G$ has prime order, contrary to assumption.