Irreducible representations of a finite group over a field are important because all representations of a group are direct sums of irreducible representations. Maschke tells us that if φ is a representation of the finite group G of order n on the m-dimensional space V over the field K of complex numbers and if U is an invariant subspace of φ, then U has a complementary reducing subspace W . The objective of this thesis is to find all irreducible representations of the dihedral group D2n. The reason we will work with the dihedral group is because it is one of the first and most intuitive non-abelian group we encounter in abstract algebra. I will compute the representations and characters of D2n and my thesis will be an explanation of these computations. When n = 2k + 1 we will show that there are k + 2 irreducible representations of D2n, but when n = 2k we will see that D2n has k + 3 irreducible rep- resentations. To achieve this we will first give some background in group, ring, module, and vector space theory that is used in representation theory. We will then explain what general representation theory is. Finally we will show how we arrived at our conclusion.

For a finite group G = {g_0 = 1, g_1,. . ., g_{n-1}} , we can associate independent variables x_0, x_1, . . ., x_{n-1} where x_i = x_{g_i}. There is a natural action of Aut(G) on C[x_0, . . . ,x_{n-})]. Let C_1, . . . , C_r be the conjugacy classes of G. If C = {g_{i_1}, g_{i_2}, . . . , g_{i_u }} is a conjugacy class, then let x(C) = x_{i_1} + x_{i_2} + . . . + x_{i_u}. Let ρG be the representation of Aut(G) on C[x_0, . . . , x_(n-1)]/〈x(C_1), . . . , x(C_r) 〉 and let Χ_G be the character afforded by ρ_G. If G is a dihedral group of the form D_2p, D_4p or D_{2p^2}, with p an odd prime, I show how Χ_G splits into irreducible constituents. I also show how the module C[x_0, . . . ,x_{n-1}]/ decomposes into irreducible submodules. This problem is motivated by results of Humphries [2] relating to random walks on groups and the group determinant.

Cryptography is defined to be the practice and studying of hiding information and is used in applications present today; examples include the security of ATM cards and computer passwords ([34]). In order to transform information to make it unreadable, one needs a series of algorithms. Many of these algorithms are based on elliptic curves because they require fewer bits. To use such algorithms, one must find the rational points on an elliptic curve. The study of Algebraic Number Theory, and in particular, rare objects known as power bases, help determine what these rational points are. With such broad applications, studying power bases is an interesting topic with many research opportunities, one of which is given below. There are many similarities between Cyclic and Dihedral fields of prime degree; more specifically, the structure of their field discriminants is comparable. Since the existence of power bases (i.e. monogenicity) is based upon finding solutions to the index form equation - an equation dependant on field discriminants - does this imply monogenic properties of such fields are also analogous? For instance, in [14], Marie-Nicole Gras has shown there is only one monogenic cyclic field of degree 5. Is there a similar result for dihedral fields of degree 5? The purpose of this thesis is to show that there exist infinitely many monogenic dihedral quintic fields and hence, not just one or finitely many. We do so by using a well- known family of quintic polynomials with Galois group D₅. Thus, the main theorem given in this thesis will confirm that monogenic properties between cyclic and dihedral quintic fields are not always correlative.

Cryptography is defined to be the practice and studying of hiding information and is used in applications present today; examples include the security of ATM cards and computer passwords ([34]). In order to transform information to make it unreadable, one needs a series of algorithms. Many of these algorithms are based on elliptic curves because they require fewer bits. To use such algorithms, one must find the rational points on an elliptic curve. The study of Algebraic Number Theory, and in particular, rare objects known as power bases, help determine what these rational points are. With such broad applications, studying power bases is an interesting topic with many research opportunities, one of which is given below. There are many similarities between Cyclic and Dihedral fields of prime degree; more specifically, the structure of their field discriminants is comparable. Since the existence of power bases (i.e. monogenicity) is based upon finding solutions to the index form equation - an equation dependant on field discriminants - does this imply monogenic properties of such fields are also analogous? For instance, in [14], Marie-Nicole Gras has shown there is only one monogenic cyclic field of degree 5. Is there a similar result for dihedral fields of degree 5? The purpose of this thesis is to show that there exist infinitely many monogenic dihedral quintic fields and hence, not just one or finitely many. We do so by using a well- known family of quintic polynomials with Galois group D₅. Thus, the main theorem given in this thesis will confirm that monogenic properties between cyclic and dihedral quintic fields are not always correlative.

Cryptography is defined to be the practice and studying of hiding information and is used in applications present today; examples include the security of ATM cards and computer passwords ([34]). In order to transform information to make it unreadable, one needs a series of algorithms. Many of these algorithms are based on elliptic curves because they require fewer bits. To use such algorithms, one must find the rational points on an elliptic curve. The study of Algebraic Number Theory, and in particular, rare objects known as power bases, help determine what these rational points are. With such broad applications, studying power bases is an interesting topic with many research opportunities, one of which is given below. There are many similarities between Cyclic and Dihedral fields of prime degree; more specifically, the structure of their field discriminants is comparable. Since the existence of power bases (i.e. monogenicity) is based upon finding solutions to the index form equation - an equation dependant on field discriminants - does this imply monogenic properties of such fields are also analogous? For instance, in [14], Marie-Nicole Gras has shown there is only one monogenic cyclic field of degree 5. Is there a similar result for dihedral fields of degree 5? The purpose of this thesis is to show that there exist infinitely many monogenic dihedral quintic fields and hence, not just one or finitely many. We do so by using a well- known family of quintic polynomials with Galois group D₅. Thus, the main theorem given in this thesis will confirm that monogenic properties between cyclic and dihedral quintic fields are not always correlative. / Graduate Studies, College of (Okanagan) / Graduate

>Magister Scientiae - MSc / The pursuit of graphs which are vertex-transitive and non-Cayley on groups has been ongoing for some time. There has long been evidence to suggest that such graphs are a very rarety in occurrence. Much success has been had in this regard with various approaches being used. The aim of this thesis is to find such a class of graphs. We will take an algebraic approach. We will define Cayley graphs on loops, these loops necessarily not being groups. Specifically, we will define meta-Cayley graphs, which are vertex-transitive by construction. The loops in question are defined as the semi-direct product of groups, one of the groups being Z₂ consistently, the other being in the class of dihedral groups. In order to prove non-Cayleyness on groups, we will need to fully determine the automorphism groups of these graphs. Determining the automorphism groups is at the crux of the matter. Once these groups are determined, we may then apply Sabidussi's theorem. The theorem states that a graph is Cayley on groups if and only if its automorphism group contains a subgroup which acts regularly on its vertex set. / Chemicals Industries Education and Training Authority (CHIETA)

In 1935, Philip Hall developed a formula for finding the number of ways of generating the group of symmetries of the icosahedron from a given number of its elements. In doing so, he defined a generalized Eulerian function. This thesis uses Hall's generalized Eulerian function to calculate generalized Eulerian functions for specific groups, namely: cyclic groups, dihedral groups, and p- groups.

Notre méthode se fonde sur le relevé, tâche habituelle de l'architecte, ici orienté comme un outil d'analyse différé de l'élaboration des formes architecturales. Le relevé engage la géométrie, la perception de celui qui le mène (on devine l'importance de cette interaction), l'histoire et le lieu dans lesquels s'inscrit l'objet étudié. A travers cinq études de cas principales : - Galeries Saint-Hubert de Bruxelles, architecte J.-P. Cluysenaar ; - Pavillon italien pour la biennale de Venise, architecte A. Anselmi ; - Hall Finlandia, Helsinki, architecte A. Aalto ; - Chapelle Sogn Benedetg, Sumvitg, architecte P. Zumthor ; - Pienza, architecte B. Rossellino, d'une part, nous mettons à jour des particularités locales ; d'autre part se crée un fil conducteur général capable de relier l'échantillon disparate des projets. Sous les contraintes géométriques et topologiques ad hoc, le schéma provisoire ainsi que le nombre de mesures nécessitées par le relevé rendent compte d'une cohérence interne des formes étudiées. Cet enseignement sur la forme vient compléter la théorie des proportions.

Let G be a finite group and V be an orthogonal four-dimensional real representation space of G where the action of G is non-free. We give necessary and sufficient conditions for the existence of a G-equivariant vector field on the representation sphere of V in the cases G is the dihedral group, the generalized quaternion group and the semidihedral group in terms of decomposition of V into irreducible representations. In the case G is abelian, where the solution is already known, we give a more elementary solution.

This thesis arose from a desire to better understand the structures of automorphism groups and inner automorphism groups of quandles. We compute and give the structure of the automorphism groups of all dihedral quandles. In their paper Matrices and Finite Quandles, Ho and Nelson found all quandles (up to isomorphism) of orders 3, 4, and 5 and determined their automorphism groups. Here we find the automorphism groups of all quandles of orders 6 and 7. There are, up to isomoprhism, 73 quandles of order 6 and 289 quandles of order 7.