Ranks And Subdegrees Of The Cyclic Group, The Dihedral Group And The Affine Group And Associated Suborbital Graphs

ABSTRACT

For a period spanning 50 years, research on the ranks, subdegrees and properties of suborbital

graphs of various groups e.g. Sn , PSL(2, ), PSL(2,q) and PGL(2,q) has drawn the attention

of several mathematicians. In this study, we find the ranks and subdegrees of the actions of

the cyclic group, the dihedral group and the affine group on some given sets. In addition, the

properties of suborbital graphs corresponding to these actions are examined. To do these, we

adopt a method that relies on the Cauchy-Frobenius Lemma. It is shown that n C and n D are

transitive on (r ) X if and only if r 1, n 1 or n. On primitivity, n C and n D are proved to be

primitive whenever n is prime and imprimitive otherwise. On the actions of n C and n D on

( ), r X where r 1 or n 1, it is found that the subdegrees and rank of n C are 1,1,…,1 n times

and n respectively while those of n D are 1,2,2,2,…,2 and

1

2

n 

or 1,1,2,2,2,…,2 and

2

2

n 

when n is odd or even respectively. For both n C and n D on ( ), r X we have formulated the

conditions for all corresponding suborbital graphs to be undirected and connected. We have

also shown that the number of connected components is gcd (n,i 1).The number of

connected suborbital graphs corresponding to the actions of n C and n D is shown to be (n)

and

1

( )

2

 n respectively. On the actions of n C and n D on [ ], r X it is proved that n C is

transitive if and only if r 1 and n D is transitive only if n  3 and r  2 and as such, 3 D is

of rank 6 and all the associated suborbital graphs are disconnected with some directed and

others undirected. Finally, we have proved that the Aff (A) acts transitively, doublytransitively

and primitively on GF(q). The subdegrees of Aff (A) are 1, q 1 thus the rank is

2. The only non-trivial suborbital graph corresponding to this action is the complete graph.