Five Free-order Quasi-primitive Permutation Groups And Related Three-degree Symmetry Maps

This thesis is devoted to study quasiprimitive permutation groups of fifth-free order and the relative cubic symmetric graphs. We notice that a group G is called of order dth-free, with d a positive integer, if there is no prime p such that pd ||G|.A well-known Cayley theorem tells us that each finite group is isomorphic to a permutation group （its right regular representation）. This theorem establishes the important position of permutation groups in modern mathematics. In recent years,based on the O’Nan-Scott-Praeger theorem （see [30]） and the maximal factoriza-tions of almost simple groups obtained by Liebeck, Praeger and Saxl （see [24]）- the research of permutation groups has been developed rapidly and succesifuly applied in many other fields （especially in the field of algebraic graph theory）.In 2005, Dietrich and Erick [8] studied the properties of finite groups of cube-free order. Using the obtained result, Li and Qiao [35] completely determined the structure of such groups, and in a subsequent paper [21], obtained some nice struc-ture properties of groups of fourth-free order. In particular, they determined all simple groups of order cube- or fourth-free order. One of the main purpose of this thesis is to study the properties of groups with fifth-free order, and in particular,classifies quasiprimitive permutation groups of fifth-free order and determines all nonabelian simple groups of fifth-free order. Notice that a transitive permutation group is called quasiprimitive if each of its minimal normal subgroups is transitive.in 1938, Fruchet proved that each permutation group can be an automorphism group of a graph, thus made a tight relation between the permutation group theory and algebraic graph theory. The second aim of this thesis is to determine cubic arc-transitive graphs Γ admitting a vertex-primitive automorphism groups of fifth-free order. In precise, we proved that one of the following holds: Γ= K₄, K₄ - 4K₂,Peterson graph O₂, or Γ is a coset graph of PSL（2,p）,where p ≡ ±1 （mod 16） is a prime.