Ranks, subdegrees and suborbital graphs of finite permutation groups

Abstract

Recent research has seen the emergence of some algebraic structures through blending of group theory, combinatorics and graph theory. The structure (G, X) is a transitive group G acting on a set X. The concepts of rank, subdegrees and suborbital graphs of (G, X) have formed a subject of recent study through variations of G and X. Several studies have taken into account the action of various subgroups of the modular group on the set of rationals including infinity ( ̂ ). Recently the action of the symmetric group Sn on various sets has been thoroughly worked on in relation to ranks, subdegrees and suborbital graphs. However, not much has been done on the action of the subgroups of Sn. In view of this, the study focused on the action of the dihedral group ( ) and the cyclic group Cn=<(12…n)> on unordered and ordered -element subsets X={1, 2, …, n}. Each of the actions on unordered subsets has been proved transitive, if and only if r=1, r=n-1 or r=n. This was determined by using the orbit-stabilizer theorem and Cauchy-Frobenius lemma on each action, G on X, under consideration. The rank of Cn on X was shown to be n, while that of Dn on X was (n+1)/2, when n is even and (n+2)/2 when n is odd. This was acheived by applying Cauchy Frobenius lemma on the action of the stabilizer of x on X to count the number of orbits of under the action of the stabilizer. The subdegrees were then deduced by counting the elements of each suborbit, by analyzing the action of the stabilizer on X. Sim’s theory was then employed to construct suborbital graphs corresponding to the actions. The construction realized 3 graphs whose properties have been discussed. The study also examined the action of a cyclic subgroup of the projective special linear group on finite subsets of the set of integers, p (integers modulo p), where the action was proved transitive, rank was shown to be p and 1 connected graph was constructed. The ranks and subdegrees are significant in determination of distance-transitive representations of the linear groups and also in characterization of rank 3 permutation groups. Some group-theoretical properties are also studied through suborbital graphs. The choice of finite sets will familiarize aspiring researchers in the subject. The results have been used to investigate primitivity of the groups, which offers an opening for further research. It is expected that the results will also provide a tool for studies in Category theory, Structure and bonding in Chemistry, Hadamard matrices and Data structures in Computer science.