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.
, K & RUTTO, M (2021). Ranks And Subdegrees Of The Cyclic Group, The Dihedral Group And The Affine Group And Associated Suborbital Graphs. Afribary. Retrieved from https://track.afribary.com/works/ranks-and-subdegrees-of-the-cyclic-group-the-dihedral-group-and-the-affine-group-and-associated-suborbital-graphs
, KANGOGO and MOSES RUTTO "Ranks And Subdegrees Of The Cyclic Group, The Dihedral Group And The Affine Group And Associated Suborbital Graphs" Afribary. Afribary, 28 May. 2021, https://track.afribary.com/works/ranks-and-subdegrees-of-the-cyclic-group-the-dihedral-group-and-the-affine-group-and-associated-suborbital-graphs. Accessed 23 Nov. 2024.
, KANGOGO, MOSES RUTTO . "Ranks And Subdegrees Of The Cyclic Group, The Dihedral Group And The Affine Group And Associated Suborbital Graphs". Afribary, Afribary, 28 May. 2021. Web. 23 Nov. 2024. < https://track.afribary.com/works/ranks-and-subdegrees-of-the-cyclic-group-the-dihedral-group-and-the-affine-group-and-associated-suborbital-graphs >.
, KANGOGO and RUTTO, MOSES . "Ranks And Subdegrees Of The Cyclic Group, The Dihedral Group And The Affine Group And Associated Suborbital Graphs" Afribary (2021). Accessed November 23, 2024. https://track.afribary.com/works/ranks-and-subdegrees-of-the-cyclic-group-the-dihedral-group-and-the-affine-group-and-associated-suborbital-graphs