Resolvable Mendelsohn designs and finite Frobenius groups

We prove the existence and give constructions of a $(p(k)-1)$-fold perfect resolvable $(v, k, 1)$-Mendelsohn design for any integers $v > k \ge 2$ with $v \equiv 1 \mod k$ such that there exists a finite Frobenius group whose kernel $K$ has order $v$ and whose complement contains an element $\phi$ of order $k$, where $p(k)$ is the least prime factor of $k$. Such a design admits $K \rtimes \langle \phi \rangle$ as a group of automorphisms and is perfect when $k$ is a prime. As an application we prove that for any integer $v = p_{1}^{e_1} \ldots p_{t}^{e_t} \ge 3$ in prime factorization, and any prime $k$ dividing $p_{i}^{e_i} - 1$ for $1 \le i \le t$, there exists a resolvable perfect $(v, k, 1)$-Mendelsohn design that admits a Frobenius group as a group of automorphisms. We also prove that, if $k$ is even and divides $p_{i} - 1$ for $1 \le i \le t$, then there are at least $\varphi(k)^t$ resolvable $(v, k, 1)$-Mendelsohn designs that admit a Frobenius group as a group of automorphisms, where $\varphi$ is Euler's totient function.Comment: Final versio

Among graphs, groups, and latin squares

A latin square of order n is an n × n array in which each row and each column contains each of the numbers {1, 2, . . . , n}. A k-plex in a latin square is a collection of entries which intersects each row and column k times and contains k copies of each symbol. This thesis studies the existence of k-plexes and approximations of k-plexes in latin squares, paying particular attention to latin squares which correspond to multiplication tables of groups. The most commonly studied class of k-plex is the 1-plex, better known as a transversal. Although many latin squares do not have transversals, Brualdi conjectured that every latin square has a near transversal—i.e. a collection of entries with distinct symbols which in- tersects all but one row and all but one column. Our first main result confirms Brualdi’s conjecture in the special case of group-based latin squares. Then, using a well-known equivalence between edge-colorings of complete bipartite graphs and latin squares, we introduce Hamilton 2-plexes. We conjecture that every latin square of order n ≥ 5 has a Hamilton 2-plex and provide a range of evidence for this conjecture. In particular, we confirm our conjecture computationally for n ≤ 8 and show that a suitable analogue of Hamilton 2-plexes always occur in n × n arrays with no symbol appearing more than n/√96 times. To study Hamilton 2-plexes in group-based latin squares, we generalize the notion of harmonious groups to what we call H2-harmonious groups. Our second main result classifies all H2-harmonious abelian groups. The last part of the thesis formalizes an idea which first appeared in a paper of Cameron and Wanless: a (k,l)-plex is a collection of entries which intersects each row and column k times and contains at most l copies of each symbol. We demonstrate the existence of (k, 4k)-plexes in all latin squares and (k, k + 1)-plexes in sufficiently large latin squares. We also find analogues of these theorems for Hamilton 2-plexes, including our third main result: every sufficiently large latin square has a Hamilton (2,3)-plex

Cayley-Dickson Loops

In this dissertation we study the Cayley-Dickson loops, multiplicative structures arising from the standard Cayley-Dickson doubling process. More precisely, the Cayley-Dickson loop Qn is the multiplicative closure of basic elements of the algebra constructed by n applications of the doubling process (the first few examples of such algebras are real numbers, complex numbers, quaternions, octonions, sedenions). Starting at the octonions, Cayley-Dickson algebras and loops become nonassociative, which presents a significant challenge in their study. We begin by describing basic properties of the Cayley–Dickson loops Qn. We establish or recall elementary facts about Qn, e.g., inverses, conjugates, orders of elements, and diassociativity. We then discuss some important subloops of Qn, for instance, associator subloop, derived subloop, nuclei, center, and show that Qn are Hamiltonian. We study the structure of the automorphism groups of Qn. We show that all subloops of Qn of order 16 fall into two isomorphism classes, in particular, any such subloop is either isomorphic to the octonion loop O16, or the quasioctonion loop O16. This helps to establish that starting at the sedenion loop, the group Aut (Qn) is isomorphic to Aut (O16) x (Z2) n−3 . Next we study two notions that are of interest in loop theory, the inner mapping group Inn(Qn) and the multiplication group Mlt(Qn). We prove that Inn(Qn) is an elementary abelian 2-group of order 22 n−2 , moreover, every f \u3e Inn(Q) is a product of disjoint transpositions of the form (x,−x). This implies that nonassociative Cayley–Dickson loops are not automorphic. The elements of Mlt(Qn) are even permutations and have order 1, 2 or 4. We show that Mlt(Qn) is a semidirect product of Inn(Qn) x Z2 and an elementary abelian 2-group K, and construct an isomorphic copy of Mlt(Qn) as an external semidirect product of two abstract elementary abelian 2-groups. The groups Innl(Qn) and Innr(Qn) are proved to be equal, elementary abelian 2-groups of order 22 n−1−1 . We also establish that Mltl(Qn) is a semidirect product of Innl(Qn) x Z2 and K, and that Mltl(Qn) and Mltr(Qn) are isomorphic. We describe basic properties of the Cayley-Dickson loops Qn, e.g., inverses, conjugates, orders of elements, and diassociativity. We discuss some important subloops of Qn, for instance, associator subloop, derived subloop, nuclei, center, and show that Qn are Hamiltonian. We show that all subloops of Qn of order 16 fall into two isomorphism classes, in particular, any such subloop is either isomorphic to the octonion loop, or the quasioctonion loop. We discuss automorphism groups, inner mapping groups, and multiplication groups of the Cayley-Dickson loops, and describe the progress made on the study of their subloop structure. We also provide incidence tetrahedra for the sedenion loop and other subloops of order 32, generalizing the idea of the octonion multiplication Fano plane