| dc.description.abstract |
The main aim of this research was to determine transitivity, primitivity, ranks, subdegrees,
and suborbital graphs of cyclic group Cn and dihedral group Dn acting on
vertices of a regular n− gon. These areas have not received much attention, in fact
most of the researchers have been focused on testing whether the action of specific
degrees of the dihedral group are primitive or transitive on the vertices of a regular n−
gon. This research extends the work of Hamma to the general degree n for both Cn
and Dn. With regard to the suborbital graphs of these two groups, nothing appears in
literature and so to some extent the results obtained in this research can be regarded as
new. In this research it has been shown that Cn and Dn act transitively on the vertices
of a regular n− gon . Also Cn and Dn act imprimitivily on the vertices of a regular n−
gon if n is not prime. The rank of Cn is shown to be n and the rank of Dn is shown
to be n
2 +1 when n is even and n+1
2 when n is odd. It is also shown that the suborbits
of Cn are not all selfpaired; only 2 are selfpaired when n is even and 1 when n is odd,
the rest are paired with each other such that △i of Cn is paired with △n−i , but all the
suborbits of Dn are selfpaired. The subdegrees of Cn are shown to be all singletons,
and the subdegrees of Dn are shown to be 1,1,2,2, · · · ,( n
2 −1) twos when n is even and
1,2,2,2, · · · ,( n−1
2 ) twos when n is odd. Further it is shown that for a suborbital Oi−1 in
Cn, (a,b) ∈ Oi−1 if and only if |b−a|=
i−1 i f b > a
n−(i−1) i f a > b
, and that all suborbital
graphs of Cn are connected if and only if n is prime. The suborbitals of Dn are shown
to be union of the paired suborbitals of Cn, and the corresponding suborbital graphs are
connected if and only if n is prime. Finally it is shown that the number of components
of the suborbital graph Gi−1 for both groups is d = gcd(n, i−1) and its girth is r = n
d ,
when d 6= n
2 and zero if d = n
2 . |
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