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Nilpotent Algebras with Maximal Class in Congruence Modular Varieties Nilpotent Algebras with Maximal Class in Congruence Modular Varieties by Xuebin Zhang A thesis submitted to the FACULTY OF GRADUATE STUDIES of the UNIVERSITY OF MANITOBA in partial fulfillment of the requirements of the drgree of Doctor of Philosophy Departme n t of Mathematics and Asuonomy University of Manitoba Winnipeg. Manitoba O August, 1998 National Library Bibliothèque nationale du Canada Acquisions and Acquisitions et Bibliographie Services services bibliographiques 395 Weihgtm Street 395, tue Weilington 0t~waONK1AW OtiawaûN KfAOiU4 Canada Canada The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distriiute or sell reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la fome de microfiche/lnlm, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts ffom it Ni la thèse ni des extraits substantiels may be printed or othewise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. THE UMVERSITY OF MANITOBA FACULTY OF GRADUATE STUDIES ***** COPYRIGHT PERiMISSION PAGE: rie Cû?GEüENCE HODULAR VARIETIES A ThesidPracticum submitted to the FacuIty of Graduate Studies of The University of Manitoba in partial fuifilent of the requirements of the degree of DOCTOB OF PE1W)SOPHY Permission has been granted to the Library of The University of Manitoba to lend or sell copies of this thesislpracticum, to the National Library of Canada to microfh this thesis and to lend or sel1 copies of the Nm, and to Dissertations Abstracts International to publish an abstract of this thesis/practicum. The author reserves other publication rights, and neither this thesis/practicum nor extensive extracts from it may be printed or otherwise reproduced without the author's written permission. Acknowledgrnents 1 am deeply grateful to my supervisor Professor Robert Quackenbush. He awakened in me an interest in Universal AIgebra; He suggested to me to investigate quatemary Mendelsohn quasigroups, and this investigation has fimdly Ied to this thesis. He patiently discussed with me my ideas, and provided me with his rxperience. 1 would like to thank the Department of Mathematics and Astronomy, especidiy Professor Lynn Battm and Professor Nathan Mendelsohn for their support and encouragement 1 appreciate the tinancial support given to me by the Faculty of Gnduate Studies of the University of Manitoba, by the Departmant of Mathematics and Asuonomy and by Professor Robert Quackenbush. A bstract Generai representation theory by polynomials for nilpotent algebns in modular congruence varietirs has been established and studied by sevenl authors. An intereshg open question is how to recursively constmct nilpotent aigebns with maximal class. In this thesis. severai algebras are investigated: quatemary Mendelsohn quasigroups. Steiner quasigroups. Steiner loops. Steiner skeins. and p-groups. First. we obtain the structure theorem of finite nilpotent quaternary Mendelsohn quasigroups. By recursive construction we prove that for any natural number n. (i) there exist a nilpotent quaternary Mendelsohn quasigroup of order 4"+' with maximal class n. and a solvahk quavmary Mendelsohn quasigroup of ordrr 4" with maximal class n; (ii) there exist a nilpornt Steiner quasigroup of ordrr 3"+l with maximal class n. and a nilpotent Steiner loop of order 2n+2with maximal class n; (iii) any subdirectly irreducible quaternary Mendelsohn quasigroup (Steiner quasigroup) of class n+ 1 with some conditions cm be expanded to a subdirectly irreducible quaternary Mendelsohn quasigroup (Steiner quasigroup) of class n+2. We aLso give a new and simple recursive construction for a nilpotent Steiner skeins of order 2n+2,th maximal class n such that di its denved Steiner loops are of nilpotence ciass n; and we tepment by polynomials the dihedral group and the generalized quaternion group of order 2"+' which are nilpoient p-groups with maximal class n. Table of Contents Chapter 1. Introduction A generai representation theory for nilpotent algebras in modular congruence varieties was established by Freese and McKenzie [7]. By applyinp this representation theory to some special algehns based on vector spaces, Guelzow [Z] obtained some generd representation theories by polynomials. Guelzow also posed an interesthg question, that is, are there any recursive constructions for finitr. nilpotent algehras with maximal class? In this thesis. several aigebras ;ue investigated: quatemary Mendelsohn quasigroups, Steiner quasigroups. Steiner loops, Steiner skeins and p-groups. In Chapter 2. we introduce basic concepts from group theory to universal algebn. and the gened representation theory of F~eseand McKenzie. In Chapter 3, we smdy quatemary Mendelsohn quasigroups. In 197 1. Mendelsohn generalized Steiner triple systems to cyclic triple systems; later such cyclic triple systems were called Mendelsohn triple systems. Similar to Steiner quasigmups which are obtained from the CO-ordinatizationof Steiner triple systems. Mendelsohn quasigroups are obtained from the CO-ordinatization of Mendelsohn triple sysams. Recently, Quackenbush introduced quatemary Mendelsohn triple systems and their corresponding quatemary Mendelsohn quasigroups. We study some algbnic pmprrties with universal algeebra as a tool; we obtain the stnichire theorem of €mite nilpotent quatemary Mendelsohn quasigroups ; and hy recursive construction we prove that for any natural numher n: (i) thex exists a nilpotent quatemary Mendelsohn quasigroup of order 4n+1 with maximal class n; (ii) there exists a solvable quatemary Mendelsohn quasigroup of order 4" with maximal class n; (üi) any subdirectly irreducible nilpotent quatemary Mendelsohn quasigroup of class n+l with some conditions can be expanded to a subdirectly irreducible nilpotent quatemary Mendelsohn quasigroup of class n+2. In Chapter 4. we investigate recursive constructions of nilpotent Steiner quasigroups and nilpotent Steiner loops. Guelzow presented a strengthened version of the representation theorem given by Klosxk [ 151 for fmite distributive Steiner quasigroups. and generalized the theorern to the class of dl finite nilpotent Steiner quasigroups and posed the following open questions: (i) Are there any recursivt: constructions for nilpotent Steiner quasigroups that raise the nilpatence class? (ü) Are the= any such constmctions for distributive Steiner quasigroups? For the fitquestion we give a recursive construction; the second question is su1 open. A similar result is given for Steiner loops. That is. (i) There exists a nilpotent Steiner quasigroup of order Y+' with maximal class n. (ii) There axists a nilpotent Steiner loop of order 2n+2with maximal class n. (iii) Any subdirectly irreducible nilpotent Steiner quasigroup of class n+ 1 can expanded to a subdirectly irreducible nilpotent Steiner quasigroup of class n+2. In Chapar 5. we give a recursive construction for a nilpotent Steiner skein of order 2n+2with maximal class n such that dl its derived Steiner loops are of nilpotence class n. Armanious and Guelzow [1,3] obtained the structure theorem of finite nilpotent Steiner skeins. Guelzow [3] gave a construction of a Steiner skein of nilpotence class n with ail derived Steiner loops of nilpotence class 1. Armanious [ 1J gave a construction for Steiner skeins of nilpotence class n with ai1 its hrived Steiner loops of nilpotence class n. In this chapter we survey the main mdts on nilpotent Steiner skrins and give a new and simple construction. in the form of polynomials. for Steiner skrins of nilpotence class n with all its derived Steiner loops of nilpotence class n. In Chapter 6. we investigate recursive constructions of finite p-groups. Some general representation throrems for finite nilpotent goups and some examples for pgroups with small order have been given by Guelzow. By ~cursivi:construction we shail represent the dihedral group and the genenlized quaternion group of order 2"+' which are nilpotent pgroups with maximal class n. At the same time. we xe that the representation theories by polynomials provide a powerful tool for isomorphism classification of finite pgroups. Chapter 2. Preliminaries 1. Groups In the theory of groups, the important concepts of Atelian (commutative) group. the center of a group, the centralizer of a normal subgroup, solvable group, and nilpotent group cmal1 be defined in tems of the commutator opention. that is, [x, y] = x'ly-'xy. Altematively. these concepts can be defined in terms of the operation [M, NJ = Sg(([x. y] : x E M, y E NI), on normal subgroups, also called the commutator (SEO()is the sub- group genented by X). Analogous conceprs. based on the multiplication of ideals, are important in ring thzory. An extension of these concepts to algebras other than groups and rings is behind one of the most exciting new directions of rexarch in pneral algebn. The following detinitions and results can be found in [2 1,221. Definition 1.1. An algcbra G = ( G; 0, -'.e ) is a group if it satisfies the following identities : (i) (xmy)ez = x.(y.z); (ii) xe = e.x = x; (iii) x.x-' = x -1 ex = e. A group G is called abelian if the commutative identity holds: (iv) x.y = y.x A group G is called qckc if G is generated hy one element. Definition 1.2. A subgroup N of a group G is normal, if g~g-'= N for every g E G. Definition 1.3. If a, b E G, the commutator of a and b, denoted by [a, b], is [a. b] = a%"nb. Definition 1.4. The commutator subgroup (or derived subgroup ) of G. denoted by G'. is the subgmup of G genented by al1 the cornmutators. The higher commutator subgroups of G are defined inductively: @O' = G, ~(i+"= ~(i)'.* that is. G('+')is the comrnutator subgroup of G? Definition 1.5. A group G is solvable if G'")= {e) for some n.
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