It has long been known that there exist finite connected tetravalent arc-transitive graphs with arbitrarily large vertex-stabilisers. However, beside a well known family of exceptional graphs, related to the lexicographic product of a cycle with an edgeless graph on two vertices, only a few such infinite families of graphs are known. In this paper, we present two more families of tetravalent arc-transitive graphs with large vertex-stabilisers, each significant for its own reason
COBISS.SI-ID: 15870553
Let G be a transitive permutation group on a set X. In the paper, we discuss s a relationship between the structures of the vertex-stabiliser G_v of a point v in X and the permutation group L_v induced by G_v on one of its orbits. It had been known that in the case where G is primitive and G_v is finite every every composition factor of the group G_v is also a section of L_v. In this paper we generalize this result to possibly imprimitive permutation groups G with infinite vertex-stabilisers, subject to certain restrictions regarding the natural permutation topology on Sym(X).
COBISS.SI-ID: 15861081
We discuss a possible approach to the study of finite arc-transitive digraphs and prove an upper bound on the order of a vertex-stabiliser in locally cyclic arc-transitive digraphs of prime out-valence.
COBISS.SI-ID: 15680601
We prove that if G is PF-group of finite exponent, then the exponent of the second homology group H_2(G, M) divides the exponent of G for every profinite trivial [[ZG]]-module M. We introduce the notion of the exponential rank of a pro-p group, and find a bound for the exponential rank of a PF-group.
COBISS.SI-ID: 15098201
In 1987 Lubotzky and Mann introduced the concept of a powerful p-group: In this paper we generalise this to powerful actions. The paper contains many fundamental properties of powerful actions. Finally, as an application,we study the non-Abelian tensor product of powerful p-groups.
COBISS.SI-ID: 15596121