Computation of Lengths of Orbits of a Subgroup in a Transitive Permutation Group

Abstract

Let (G, Ω) be a transitive permutation group and F be a subgroup of G. We denote by n 1,n 2,…, n r the lengths of the orbits of F in its action on Ω. In the present paper a method for the description of (F, Ω) as a permutation group and, in particular, for the computation of the numbers n i , 1 ≤i ≤r, is given1). To realize this method, certain information about the structure of the group G is necessary. Knowledge of the length of orbits of certain subgroups in a transitive permutation group enables us to effectively investigate this group as well as certain combinatorial objects which admit this group as an automorphism group. For example, in Section 3 this method is used to construct a new cubic graph on 110 vertices which is edge- but not vertex-transitive and which admits PGL 2 (11) as automorphism group.KeywordsAutomorphism GroupConjugacy ClassPermutation GroupIrreducible CharacterFrobenius GroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.