Sylow's theorem for Moufang loops

For finite Moufang loops, we prove an analog of the first Sylow theorem giving a criterion of the existence of a p-Sylow subloop. We also find the maximal order of p-subloops in the Moufang loops that do not possess p-Sylow subloops.Comment: 21 page

Code loops in dimension at most 8

Code loops are certain Moufang $2$-loops constructed from doubly even binary codes that play an important role in the construction of local subgroups of sporadic groups. More precisely, code loops are central extensions of the group of order $2$ by an elementary abelian $2$-group $V$ in the variety of loops such that their squaring map, commutator map and associator map are related by combinatorial polarization and the associator map is a trilinear alternating form. Using existing classifications of trilinear alternating forms over the field of $2$ elements, we enumerate code loops of dimension $d=\mathrm{dim}(V)\le 8$ (equivalently, of order $2^{d+1}\le 512$) up to isomorphism. There are $767$ code loops of order $128$, and $80826$ of order $256$, and $937791557$ of order $512$

Classification of Moufang Loops of Odd Order pq3.

An open problem in the theory of Moufang loops is to classify those loops which are minimally non associative, that is, loops which are non associative but where all proper subloops are associative. A related question is to classify all integers n for which a minimally nonassociative Moufang loop exists

Computable Model Theory on Loops

We give an introduction to the problem of computable algebras. Specifically, the algebras of loops and groups. We start by defining a loop and group, then give some of their properties. We then give an overview of comptability theory, and apply it to loops and groups. We conclude by showing that a finitely presented residually finite algebra has a solvable word problem

On Loop Commutators, Quaternionic Automorphic Loops, and Related Topics

This dissertation deals with three topics inside loop and quasigroup theory. First, as a continuation of the project started by David Stanovský and Petr Vojtĕchovský, we study the commutator of congruences defined by Freese and McKenzie in order to create a more pleasing, equivalent definition of the commutator inside of loops. Moreover, we show that the commutator can be characterized by the generators of the inner mapping group of the loop. We then translate these results to characterize the commutator of two normal subloops of any loop. Second, we study automorphic loops with the desire to find more examples of small orders. Here we construct a family of automorphic loops, called quaternionic automorphic loops, which have order 2n for n ≥ 3, and prove several theorems about their structure. Although quaternionic automorphic loops are nonassociative, many of their properties are reminiscent of the generalized quaternion groups. Lastly, we find varieties of quasigroups which are isotopic to commutative Moufang loops and prove their full characterization. Moreover, we define a new variety of quasigroups motivated by the semimedial quasigroups and show that they have an affine representation over commutative Moufang loops similar to the semimedial case proven by Kepka

Moufang Loops, Magmas And The Moufang Identities

Loop theory is a generalization of group theory; Moufang loops are a variety of loops. Four equivalent (Moufang) identities axiomatize these loops. Moufang loops also share many similar properties as groups though generally they are not associative; Moufang’s Theorem is pivotal in establishing this close relationship. The existing proof of the equivalence of the Moufang identities involves the notion of "autotopism", a completely difficult concept in itself, whereas there is no known complete proof of the Moufang’s Theorem (though several reasonably acceptable proofs exist). This thesis provides a simple, basic and complete proof of both. Moreover, the equivalence of the localized versions of the four identities is studied under the generalized setting of magmas and proven under necessary and sufficient conditions. Finally, this research gives a (partial) resolution of Moufang loops of odd order p2q4

Octonion Internal Space Algebra for the Standard Model

Our search for an appropriate notion of internal space for the fundamental particles starts with the Clifford algebra $C\ell_{10}$ with gamma matrices expressed as left multiplication by octonion units times a pair of Pauli matrices. Fixing an imaginary octonion unit allows to write ${\mathbb O} = {\mathbb C} \oplus {\mathbb C}^3$ reflecting the lepton-quark symmetry. We identify the preserved unit with the $C\ell_6$ pseudoscalar, $\omega_6 = \gamma_1 \cdots \gamma_6$. It is fixed by the Pati-Salam subgroup of $Spin(10)$, $G_{\rm PS} = Spin (4) \times Spin (6) / {\mathbb Z}_2$, which respects the splitting $C\ell_{10}=C\ell_4\hat{\otimes} C\ell_6$, while ${\cal P} = \frac12 (1 - i\omega_6)$ is the projector on the 16-dimensional particle subspace (annihilating the antiparticles). We express the generators of the subalgebras $C\ell_4$ and $C\ell_6$ in terms of fermionic oscillators describing flavour and colour, respectively. The standard model gauge group appears as the subgroup of $G_{PS}$ that preserves the sterile neutrino (identified with the Fock vacuum). The $\mathbb{Z}_2$-graded internal space algebra $\mathcal{A}$ is then included in the projected tensor product: $\mathcal{A}\subset \mathcal{P}C\ell_{10}\mathcal{P}=C\ell_4\otimes C\ell_6^0$. The Higgs field appears as the scalar term of a superconnection, an element of the odd part, $C\ell_4^1$, of the first factor. As an application we express the ratio $\frac{m_H}{m_W}$ of the Higgs to the $W$-boson masses in terms of the cosine of the theoretical Weinberg angle.Comment: 32 pages, Extended version of a lecture presented at the Workshop Octonions and the Standard Model, Perimeter Institute, Waterloo, Canada, February-May 2021, and at the 14th International Workshop Lie Theory and Its Applications to Physics (LT 14), Sofia, June 2021. v2: reference added, typos correcte

Dihedral-Like Constructions of Automorphic Loops

In this dissertation we study dihedral-like constructions of automorphic loops. Automorphic loops are loops in which all inner mappings are automorphisms. We start by describing a generalization of the dihedral construction for groups. Namely, if (G , +) is an abelian group, m \u3e 1 and α ∈2 Aut(G ), let Dih(m, G, α) on Zm × G be defined by (i, u )(j, v ) = (i + j , ((-1)j u + v )αij ). We prove that the resulting loop is automorphic if and only if m = 2 or (α2 = 1 and m is even) or (m is odd, α = 1 and exp(G ) ≤ 2). In the last case, the loop is a group. The case m = 2 was introduced by Kinyon, Kunen, Phillips, and Vojtěchovský. We study basic structural properties of dihedral-like automorphic loops. We describe certain subloops, including: nucleus, commutant, center, associator subloop and derived subloop. We prove theorems for dihedral-like automorphic loops analogous to the Cauchy and Lagrange theorems for groups, and further we discuss the coset decomposition in dihedral-like automorphic loops. We show that two finite dihedral-like automorphic loops Dih( m, G, α) and Dih(m̅, Ḡ, ᾱ) are isomorphic if and only if m= m̅, G [congruent with] Ḡ and α is conjugate to ᾱ in Aut(G ). We describe the automorphism group of Q and its subgroup consisting of inner mappings of Q . Finally, due to the solution to the isomorphism problem, we are interested in studying conjugacy classes of automorphism groups of finite abelian groups. Then we describe all dihedral-like automorphic loops of order \u3c 128 up to isomorphism. We conclude with a description of all dihedral-like automorphic loops of order \u3c 64 up to isotopism