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 > 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.
","abstract_html":"<p>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 <em>Q<sub>n</sub></em> is the multiplicative closure of basic elements of the algebra constructed by <em>n</em> 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.</p> <p>We begin by describing basic properties of the Cayley–Dickson loops <em>Q<sub>n</sub></em>. We establish or recall elementary facts about <em>Q<sub>n</sub></em>, e.g., inverses, conjugates, orders of elements, and diassociativity. We then discuss some important subloops of <em>Q<sub>n</sub></em>, for instance, associator subloop, derived subloop, nuclei, center, and show that <em>Q<sub>n</sub></em> are Hamiltonian. We study the structure of the automorphism groups of <em>Q<sub>n</sub></em>. We show that all subloops of <em>Q<sub>n</sub></em> of order 16 fall into two isomorphism classes, in particular, any such subloop is either isomorphic to the octonion loop O<sub>16</sub>, or the quasioctonion loop O<sub>16</sub>. This helps to establish that starting at the sedenion loop, the group Aut (<em>Q<sub>n</sub></em>) is isomorphic to Aut (O<sub>16</sub>) x (Z<sub>2</sub>) <sup>n−3</sup> .</p> <p>Next we study two notions that are of interest in loop theory, the inner mapping group <em>Inn</em>(<em>Q<sub>n</sub></em>) and the multiplication group <em>Mlt</em>(<em>Q<sub>n</sub></em>). We prove that <em>Inn</em>(<em>Q<sub>n</sub></em>) is an elementary abelian 2-group of order 2<sup>2 n−2</sup> , moreover, every <em>f</em> > <em>Inn</em>(<em>Q</em>) is a product of disjoint transpositions of the form (<em>x,−x</em>). This implies that nonassociative Cayley–Dickson loops are not automorphic. The elements of <em>Mlt</em>(<em>Q<sub>n</sub></em>) are even permutations and have order 1, 2 or 4. We show that <em>Mlt</em>(<em>Q<sub>n</sub></em>) is a semidirect product of <em>Inn</em>(<em>Q<sub>n</sub></em>) x <em>Z<sub>2</sub></em> and an elementary abelian 2-group <em>K</em>, and construct an isomorphic copy of <em>Mlt</em>(<em>Q<sub>n</sub></em>) as an external semidirect product of two abstract elementary abelian 2-groups. The groups<em> Inn</em><sub>l</sub>(<em>Q<sub>n</sub></em>) and <em>Inn<sub>r</sub></em>(<em>Q<sub>n</sub></em>) are proved to be equal, elementary abelian 2-groups of order 22 <sup>n−1−1</sup> . We also establish that <em>Mltl</em>(<em>Q<sub>n</sub></em>) is a semidirect product of<em> Inn</em><sub>l</sub>(<em>Q<sub>n</sub></em>) x Z<sub>2</sub> and <em>K</em>, and that <em>Mlt</em><sub>l</sub>(<em>Q<sub>n</sub></em>) and <em>Mlt<sub>r</sub></em>(<em>Q<sub>n</sub></em>) are isomorphic.</p> <p>We describe basic properties of the Cayley-Dickson loops <em>Q<sub>n</sub></em>, e.g., inverses, conjugates, orders of elements, and diassociativity. We discuss some important subloops of <em>Q<sub>n</sub></em>, for instance, associator subloop, derived subloop, nuclei, center, and show that <em>Q<sub>n</sub></em> are Hamiltonian. We show that all subloops of <em>Q<sub>n</sub></em> 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.</p>","abstract_has_math":false,"creators":["Kirshtein, Jenya"],"institution":null,"degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":null,"degree_department":null,"school":null,"contributors":["Petr Vojtechovsky, Ph.D.","Michael Kinyon","Richard Ball","Nikolaos Galatos","Richard Green","J. Michael Daniels"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-01-01T08:00:00Z","date_published":"2012-01-01T08:00:00Z","updated_at":"2026-07-24T02:02:30Z","subjects":["Cayley-Dickson doubling process","Loop theory","Multiplication group","Nonassociative","Octonion","Sedenion","Mathematics"],"languages":["en"],"rights":["Copyright is held by the author. User is responsible for all copyright compliance.
"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.du.edu/etd/849","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Petr Vojtechovsky, Ph.D.","Michael Kinyon","Richard Ball","Nikolaos Galatos","Richard Green","J. Michael Daniels"]},{"key":"dc:creator","label":"Author","values":["Kirshtein, Jenya"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2001-01-01T08:00:00Z"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Cayley-Dickson doubling process","Loop theory","Multiplication group","Nonassociative","Octonion","Sedenion","Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright is held by the author. User is responsible for all copyright compliance.
"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.du.edu/etd/849"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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 > 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.
"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Cayley-Dickson Loops"]}]}],"canonical_facts":{"dc:contributor":["Petr Vojtechovsky, Ph.D.","Michael Kinyon","Richard Ball","Nikolaos Galatos","Richard Green","J. Michael Daniels"],"dc:creator":["Kirshtein, Jenya"],"dc:date.available":["2001-01-01T08:00:00Z"],"dc:description.abstract":["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 > 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.
"],"dc:format":["application/pdf"],"dc:identifier":["https://digitalcommons.du.edu/etd/849"],"dc:language":["en"],"dc:rights":["Copyright is held by the author. User is responsible for all copyright compliance.
"],"dc:subject":["Cayley-Dickson doubling process","Loop theory","Multiplication group","Nonassociative","Octonion","Sedenion","Mathematics"],"dc:title":["Cayley-Dickson Loops"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."]},"updated_at":"2026-07-24T02:02:30Z"}