An O'nan-scott Theorem for Finite Quasiprimitive Permutation Groups and an Application to 2-arc Transitive Graphs
نویسنده
چکیده
A permutation group is said to be quasiprimitive if each of its nontrivial normal subgroups is transitive. A structure theorem for finite quasiprimitive permutation groups is proved, along the lines of the O'NanScott Theorem for finite primitive permutation groups. It is shown that every finite, non-bipartite, 2-arc transitive graph is a cover of a quasiprimitive 2-arc transitive graph. The structure theorem for quasiprimitive groups is used to investigate the structure of quasiprimitive 2-arc transitive graphs, and a new construction is given for a family of such graphs.
منابع مشابه
Permutation groups and derangements of odd prime order
Let G be a transitive permutation group of degree n. We say that G is 2′elusive if n is divisible by an odd prime, but G does not contain a derangement of odd prime order. In this paper we study the structure of quasiprimitive and biquasiprimitive 2′-elusive permutation groups, extending earlier work of Giudici and Xu on elusive groups. As an application, we use our results to investigate autom...
متن کاملOn automorphism groups of quasiprimitive 2-arc transitive graphs
We characterize the automorphism groups of quasiprimitive 2-arc-transitive graphs of twisted wreath product type. This is a partial solution for a problem of Praeger regarding quasiprimitive 2-arc transitive graphs. The solution stimulates several further research problems regarding automorphism groups of edge-transitive Cayley graphs and digraphs.
متن کاملPermutation groups and normal subgroups
Various descending chains of subgroups of a finite permutation group can be used to define a sequence of 'basic' permutation groups that are analogues of composition factors for abstract finite groups. Primitive groups have been the traditional choice for this purpose, but some combinatorial applications require different kinds of basic groups, such as quasiprimitive groups, that are defined by...
متن کاملTwo-geodesic transitive graphs of prime power order
In a non-complete graph $Gamma$, a vertex triple $(u,v,w)$ with $v$ adjacent to both $u$ and $w$ is called a $2$-geodesic if $uneq w$ and $u,w$ are not adjacent. The graph $Gamma$ is said to be $2$-geodesic transitive if its automorphism group is transitive on arcs, and also on 2-geodesics. We first produce a reduction theorem for the family of $2$-geodesic transitive graphs of prime power or...
متن کاملLocally s-distance transitive graphs
We give a unified approach to analysing, for each positive integer s, a class of finite connected graphs that contains all the distance transitive graphs as well as the locally s-arc transitive graphs of diameter at least s. A graph is in the class if it is connected and if, for each vertex v, the subgroup of automorphisms fixing v acts transitively on the set of vertices at distance i from v, ...
متن کامل