Bounds on finite quasiprimitive permutation groups (Q2765556)

A permutation group \(G\) on a finite set \(\Omega\) of size \(n\) is said to be quasiprimitive on \(\Omega\) if every nontrivial normal subgroup of \(G\) is transitive on \(\Omega\). Every primitive permutation group on \(\Omega\) is quasiprimitive but the converse is not true. Moreover, a quasiprimitive permutation group on \(\Omega\) is isomorphic to a primitive permutation group of degree smaller than \(n\). The authors show that several properties of primitive permutation groups remain valid for quasiprimitive permutation groups, sometimes with suitable adjustments.NEWLINENEWLINENEWLINEIn particular, they extend from primitive groups to quasiprimitive groups a series of results concerning \(p\)-cycles, \(p\)-elements, Jordan sets, upper bounds on the order, minimum base size, restricted sections, lower bounds on the minimal degree and density of the orders. Some of these results rely on the classification of finite simple groups.NEWLINENEWLINENEWLINEA number of open classification problems related to quasiprimitive groups are proposed.