Cliques in derangement graphs for innately transitive groups (Q6611928)

Let \(G\) be a permutation group of finite degree; then \(G\) is called innately transitive if it has a minimal normal subgroup which is transitive. The class of innately transitive groups contains both the primitive and quasiprimitive groups and there is a version of the O'Nan-Scott theorem which holds for these groups (see [\textit{J. Bamberg} and \textit{C. E. Praeger}, Proc. Lond. Math. Soc. (3) 89, No. 1, 71--103 (2004; Zbl 1069.20003)]). Independently, we define the derangement graph of any permutation group \(G\) to be the graph \(\Gamma _{G}\) whose vertex set is \(G\) and where two vertices \(g\) and \(h\) are joined by an edge if and only if \(gh^{-1}\) has no fixed points.\N\NThe main results of the paper are the following:\N\NTheorem 1.1. and 1.2. There exists a function \(f_{1}\) such that, if \(G\) is innately transitive of degree \(n\) and \(\Gamma _{G}\) has no clique of size \(k\), then \(n\leq f_{1}(k)\); in particular, \(f_{1}(4)=3\). \N\NTheorem 1.3. There exists a function \(f_{2}\) such that if \(G\) is innately transitive of degree \(n\) and \(G\) has no semiregular subgroup of order at least \(k\) then either \(n\leq f_{2}(k)\) or for some integer \(\kappa >0\): \begin{enumerate}\N\item \(G\) is primitive of degree \(12^{\kappa }\) \N\item \( G\) is a wreath product of the form \(M_{11}~wr~A\) where \(M_{11}\) is the Mathieu group and \(A\) is a transitive group of degree \(12\).\end{enumerate}\N\NIf \(k\leq 4\) then the structure of \(G\) is described in more detail. Theorem 1.3 is related to the poly-circulant conjecture (see [\textit{M. Giudici}, J. Lond. Math. Soc., II. Ser. 67, No. 1, 73--84 (2003; Zbl 1050.20002)]).