ABSTRACT
Many authors have studied the suborbital graphs of various group actions and their
corresponding properties. This thesis investigates the actions of the cyclic group and the
dihedral group on the diagonals of a regular -gon and the properties of their corresponding
suborbital graphs. In addition, it focuses on the action of the multiplicative group of units
on
, the set of non zero elements in . The properties of the
corresponding suborbital graphs to this action are also investigated. Lastly, graphs whose
autormorphism groups contain the cyclic group and the dihedral group are constructed.
Transitivity of the actions is established using the Cauchy- Frobenius Lemma or the Orbit-
Stabilizer Theorem. Schur’s algorithm is employed to construct all graphs whose groups of
automorphism contain the cyclic group and the dihedral group . It has been shown that,
and acts transitively on the set of diagonals of a regular - gon and the action is imprimitive
when
is not a prime. The cyclic group, , has two self-paired suborbits when
and only one self-paired suborbit when , while the dihedral group has all its
suborbits self-paired. For the two groups it has been shown that the number, , of the connected
components of the suborbital graph is
and its girth is
when , otherwise it is
zero. The action of on
is shown to be transitive if and only if is a prime.
This action is imprimitive on
when and it has suborbits with two of them being
self-paired. The number of connected components of the suborbital graph , corresponding to
the suborbital , is
. Its girth is , where is the order of the element in .
Finally, it has been shown that if is a permutation group acting transitively on a set of
cardinality and the action is of rank , then the number of regular graphs on vertices whose
groups of automorphism contain is;
, where is the number of
self-paired suborbits of . These among other results have been stated as Lemmas and
Theorems.
Olum, F (2021). Suborbital Graphs Of The Groups C, D And U And Graphs Whose Automorphism Groups Contain The Permutation Groups C And D. Afribary. Retrieved from https://afribary.com/works/suborbital-graphs-of-the-groups-c-d-and-u-and-graphs-whose-automorphism-groups-contain-the-permutation-groups-c-and-d
Olum, Fredrick "Suborbital Graphs Of The Groups C, D And U And Graphs Whose Automorphism Groups Contain The Permutation Groups C And D" Afribary. Afribary, 28 May. 2021, https://afribary.com/works/suborbital-graphs-of-the-groups-c-d-and-u-and-graphs-whose-automorphism-groups-contain-the-permutation-groups-c-and-d. Accessed 02 Nov. 2024.
Olum, Fredrick . "Suborbital Graphs Of The Groups C, D And U And Graphs Whose Automorphism Groups Contain The Permutation Groups C And D". Afribary, Afribary, 28 May. 2021. Web. 02 Nov. 2024. < https://afribary.com/works/suborbital-graphs-of-the-groups-c-d-and-u-and-graphs-whose-automorphism-groups-contain-the-permutation-groups-c-and-d >.
Olum, Fredrick . "Suborbital Graphs Of The Groups C, D And U And Graphs Whose Automorphism Groups Contain The Permutation Groups C And D" Afribary (2021). Accessed November 02, 2024. https://afribary.com/works/suborbital-graphs-of-the-groups-c-d-and-u-and-graphs-whose-automorphism-groups-contain-the-permutation-groups-c-and-d