5 research outputs found
Semiregular automorphisms of vertex-transitive graphs of certain valencies
AbstractIt is shown that a vertex-transitive graph of valency p+1, p a prime, admitting a transitive action of a {2,p}-group, has a non-identity semiregular automorphism. As a consequence, it is proved that a quartic vertex-transitive graph has a non-identity semiregular automorphism, thus giving a partial affirmative answer to the conjecture that all vertex-transitive graphs have such an automorphism and, more generally, that all 2-closed transitive permutation groups contain such an element (see [D. MaruÅ”iÄ, On vertex symmetric digraphs, Discrete Math. 36 (1981) 69ā81; P.J. Cameron (Ed.), Problems from the Fifteenth British Combinatorial Conference, Discrete Math. 167/168 (1997) 605ā615])
Minimal normal subgroups of transitive permutation groups of square-free degree
It is shown that a minimal normal subgroup of a transitive permutation group of square-free degree in its induced action is simple and quasiprimitive, with three exceptions related to ā«ā«, ā«ā«, and PSL(2,29). Moreover, it is shown that a minimal normal subgroup of a 2-closed permutation group of square-free degree in its induced action is simple. As an almost immediate consequence, it follows that a 2-closed transitive permutation group of square-free degree contains a semiregular element of prime order, thus giving a partial affirmative answer to the conjecture that all 2-closed transitive permutation groups contain such an element (see [D. MaruÅ”ic, On vertex symmetric digraphs,Discrete Math. 36 (1981) 69-81P.J. Cameron (Ed.), Problems from the fifteenth British combinatorial conference, Discrete Math. 167/168 (1997) 605-615])