Group Theory
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
General Regulations for Series Run on Circuits / Automobile Sport (As on 29.05.2020)
General Regulations for Series run on Circuits / Automobile Sport (as on 29.05.2020) Name of the Series: Tourenwagen Classics 2020 DMSB Visa Number: 631/20 Status of the Series/Events: National A Plus incl. NSAFP This series is to make the heroic appearances of the 'golden era' of touring car racing of the 70s, 80s and 90s with all their personalities and vehicles back in the race. A sporty field for private drivers and ex-professionals who want to use their technically complex racing cars in the demanding but cost-conscious setting. Only vehicles that are in their appearance from that era and are seen in those series such as DTM, DRM, STW, BTCC, ETCC or other similar series are addressed. At designated events and by invitation only, the vehicle field will be supplemented by the heroes of the German Racing Championship (formerly DRM) and other iconic cars known , in order to be able to offer organizers, fans and drivers a thrilling spectrum of motorsport history Promoter / Organisation: Tourenwagen Classics GmbH Contacts: Ralph Bahr Mobil-No.: +49 173 1644114 Michael Thier Mobil-No.: +49 171 5104881 Homepage: www.tourenwagen-classics.com E-Mail: [email protected] Table of Contents Part 1 Sporting Regulations 1. Introduction 2. Organisation 2.1 Details on titles and awards of the Series 2.2 Name of the parent ASN 2.3 ASN Visa/Registration Number 2.4 Name of the Organiser/Promoter, address and contacts (Permanent office) 2.5 Composition of the organising committee 2.6 List of Officials (Permanent Stewards) 3. Regulations and Legal Basis of the Series 3.1 Official language 3.2 Responsibility, modification of the regulations, cancellation of the event 4. -
1 Useful English/German Vocabulary 2 Isomorphism Theorems Continued
1 Useful English/German Vocabulary simple group -einfache Gruppe normal series - Normalreihe composition series - Kompositionsreihe refinement - Verfeinerung 2 Isomorphism theorems continued Theorem 2.1 (Lattice isomorphism theorem). Let G be a group and let N be a normal subgroup of G. If A is a subgroup of G containing N, let A := A=N.Let π : G ! G=N be the canonical projection. The map A 7! π(A) = A is a bijection between the set of subgroups of G containing N and the set of subgroups of G=N. Moreover, if A; B ≤ G with N ≤ A and N ≤ B then: 1. A ≤ B if and only if A ≤ B; and in this case [B : A] = [B : A] ∼ 2. A C B is and only if A C B; and in this case B=A = B=A 3. hA; Bi = hA; Bi 4. A \ B = A \ B Proof. UB9 Theorem 2.2 (Butterfly Lemma /Zassenhaus Lemma). Let aCA and b C B be subgroups of a group G. Then a(A \ b) is a normal in a(A \ B); b(B \ a) is normal in b(B \ A); (A \ b)(B \ a) is normal in (A \ B) and a(A \ B) (A \ B) b(B \ A) ∼= ∼= : a(A \ b) (A \ b)(B \ a) b(B \ a) Proof. Note first that since A ≤ NG(a) and B ≤ NG(b), we have that A \ b ≤ A \ B ≤ NG(a) and B \ a ≤ A \ B ≤ NG(b): Thus a(A \ b); a(A \ B); b(B \ a) and b(B \ A) are subgroups of G (see lecture 17 corollary 1.10). -
Octonion Multiplication and Heawood's
CONFLUENTES MATHEMATICI Bruno SÉVENNEC Octonion multiplication and Heawood’s map Tome 5, no 2 (2013), p. 71-76. <http://cml.cedram.org/item?id=CML_2013__5_2_71_0> © Les auteurs et Confluentes Mathematici, 2013. Tous droits réservés. L’accès aux articles de la revue « Confluentes Mathematici » (http://cml.cedram.org/), implique l’accord avec les condi- tions générales d’utilisation (http://cml.cedram.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation á fin strictement personnelle du copiste est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Confluentes Math. 5, 2 (2013) 71-76 OCTONION MULTIPLICATION AND HEAWOOD’S MAP BRUNO SÉVENNEC Abstract. In this note, the octonion multiplication table is recovered from a regular tesse- lation of the equilateral two timensional torus by seven hexagons, also known as Heawood’s map. Almost any article or book dealing with Cayley-Graves algebra O of octonions (to be recalled shortly) has a picture like the following Figure 0.1 representing the so-called ‘Fano plane’, which will be denoted by Π, together with some cyclic ordering on each of its ‘lines’. The Fano plane is a set of seven points, in which seven three-point subsets called ‘lines’ are specified, such that any two points are contained in a unique line, and any two lines intersect in a unique point, giving a so-called (combinatorial) projective plane [8,7]. -
A Preliminary Study of University Students' Collaborative Learning Behavior Patterns in the Context of Online Argumentation Le
A Preliminary Study of University Students’ Collaborative Learning Behavior Patterns in the Context of Online Argumentation Learning Activities: The Role of Idea-Centered Collaborative Argumentation Instruction Ying-Tien Wu, Li-Jen Wang, and Teng-Yao Cheng [email protected], [email protected], [email protected] Graduate Institute of Network Learning Technology, National Central University, Taiwan Abstract: Learners have more and more opportunities to encounter a variety of socio-scientific issues (SSIs) and they may have difficulties in collaborative argumentation on SSIs. Knowledge building is a theory about idea-centered collaborative knowledge innovation and creation. The application of idea-centered collaboration practice as emphasized in knowledge building may be helpful for facilitating students’ collaborative argumentation. To examine the perspective above, this study attempted to integrate idea-centered collaboration into argumentation practice. The participants were 48 university students and were randomly divided into experimental and control group (n=24 for both groups). The control group only received argumentation instruction, while the experimental group received explicit idea-centered collaborative argumentation (CA) instruction. This study found that two groups of students revealed different collaborative learning behavior patterns. It is also noted that the students in the experimental group benefited more in collaborative argumentation from the proper adaption of knowledge building and explicit idea-centered collaborative argumentation instruction. Introduction In the knowledge-based societies, learners have more and more opportunities to encounter a variety of social dilemmas coming with rapid development in science and technologies. These social dilemmas are often termed “Socio-scientific issues (SSIs)” which are controversial social issues that are generally ill-structured, open-ended authentic problems which have multiple solutions (Sadler, 2004; Sadler & Zeidler, 2005). -
More on the Sylow Theorems
MORE ON THE SYLOW THEOREMS 1. Introduction Several alternative proofs of the Sylow theorems are collected here. Section 2 has a proof of Sylow I by Sylow, Section 3 has a proof of Sylow I by Frobenius, and Section 4 has an extension of Sylow I and II to p-subgroups due to Sylow. Section 5 discusses some history related to the Sylow theorems and formulates (but does not prove) two extensions of Sylow III to p-subgroups, by Frobenius and Weisner. 2. Sylow I by Sylow In modern language, here is Sylow's proof that his subgroups exist. Pick a prime p dividing #G. Let P be a p-subgroup of G which is as large as possible. We call P a maximal p-subgroup. We do not yet know its size is the biggest p-power in #G. The goal is to show [G : P ] 6≡ 0 mod p, so #P is the largest p-power dividing #G. Let N = N(P ) be the normalizer of P in G. Then all the elements of p-power order in N lie in P . Indeed, any element of N with p-power order which is not in P would give a non-identity element of p-power order in N=P . Then we could take inverse images through the projection N ! N=P to find a p-subgroup inside N properly containing P , but this contradicts the maximality of P as a p-subgroup of G. Since there are no non-trivial elements of p-power order in N=P , the index [N : P ] is not divisible by p by Cauchy's theorem. -
The Sylow Theorems and Classification of Small Finite Order Groups
Union College Union | Digital Works Honors Theses Student Work 6-2015 The yS low Theorems and Classification of Small Finite Order Groups William Stearns Union College - Schenectady, NY Follow this and additional works at: https://digitalworks.union.edu/theses Part of the Physical Sciences and Mathematics Commons Recommended Citation Stearns, William, "The yS low Theorems and Classification of Small Finite Order Groups" (2015). Honors Theses. 395. https://digitalworks.union.edu/theses/395 This Open Access is brought to you for free and open access by the Student Work at Union | Digital Works. It has been accepted for inclusion in Honors Theses by an authorized administrator of Union | Digital Works. For more information, please contact [email protected]. THE SYLOW THEOREMS AND CLASSIFICATION OF SMALL FINITE ORDER GROUPS WILLIAM W. STEARNS Abstract. This thesis will provide an overview of various topics in group theory, all in order to accomplish the end goal of classifying all groups of order up to 15. An important precursor to classifying finite order groups, the Sylow Theorems illustrate what subgroups of a given group must exist, and constitute the first half of this thesis. Using these theorems in the latter sections we will classify all the possible groups of various orders up to isomorphism. In concluding this thesis, all possible distinct groups of orders up to 15 will be defined and the groundwork set for further study. 1. Introduction The results in this thesis require some background knowledge and motivation. To that end, material covered in an introductory course on abstract algebra should be sufficient. In particular, it is assumed that the reader is familiar with the concepts and definitions of: group, subgroup, coset, index, homomorphism, and kernel. -
(Group N) Component Substitution Application Form
5th Category Historic Touring Car (Group N) Component Substitution Application Form Applicant Information Full Name: Savy Dick Last First M.I. Address: 37 Adrian Rd Street Address Campbellfield Vic 3061 City State Postcode Home Phone: ( 03 ) 93576330 Mobile Phone: ( ) 0408 728 967 E-mail Address: [email protected] CAMS Licence No: COD No: Vehicle Information Make of vehicle: Ford Historic Grp.: Nc Model: Mustang / XW GT Falcon Type/identification: Year of manufacture: 1969 / 1970 Year Represents: Chassis No: Component Substitution Give a brief outline or general view of the substitution. To provide a cost effective solution for the engine block in the Mustangs and Falcons that run the 351 Windsor engine in Group Nc Historic Touring Cars. These engines were only made in small numbers compared to other V8 engines, and now almost 50 years on, they are becoming very hard (almost impossible) to find in a serviceable condition. These standard early 351Windsor engine blocks have a common problem of cracking through the main bearing tunnel webbing and the mid 90s blocks commonly crack through the lifter valley area, so after spending a great deal of time trying to find a standard block, you then need to spend even more money and time to crack test these blocks and hope you have found a good one. The engine block we are proposing is a Ford replacement block (similar to the 302 replacement engine block used in Group N already), that is readily available at a reasonable cost both here in Australia and the USA. Page | 1 Wednesday, 20 June 2018 The following Component Substitution Criteria from section 3.6.4 of the CAMS Manual of Motor Sport, 5 th Category Historic Cars must be addressed as part of the submission. -
South African Champions (Up to 2006)
SOUTH AFRICAN CHAMPIONS (UP TO 2006) CIRCUIT CARS: SA MOTOR RACING DRIVERS CHAMPIONSHIP 1953 D.H. Duff 1954 D.E. Jennings 1955 F. Brodie 1956 D.E. Jennings 1957 D.E. Jennings 1958 I.J. Fraser-Jones 1959 I.J. Fraser-Jones 1960 S. van der Vyver 1961 S. van der Vyver 1962 E. Pieterse 1963 N. Lederle 1964 J. Love 1965 J. Love 1966 J. Love 1967 J. Love 1969 J. Love 1970 D. W. Charlton 1971 D.W. Charlton 1972 D.W. Charlton 1973 D.W. Charlton 1974 D.W. Charlton 1975 D.W. Charlton 1976 I. Scheckter 1977 I. Scheckter 1978 I. Scheckter 1979 I. Scheckter 1980 T. Martin 1981 B. Tilanus 1982 G. Duxbury 1983 I. Scheckter 1984 I. Scheckter 1985 T. van Rooyen 1986 W. Taylor 1987 Discontinued SA DRIVERS CHAMPIONSHIP 1990 A. Taylor 1991 M. Briggs 1992 S. van der Linde 1993 D. Vos 1994 M. dos Santos 1995 E. van der Linde 1996 M. Jurgens 1997 J. Smith 1998 J. Smith 1999 J. Fourie 2000 Discontinued SA SALOON CAR CHAMPIONSHIP 1964 J. Swanepoel 1965 J.R. Olthoff 1966 B.V. van Rooyen 1967 B.V. van Rooyen 1968 A.W. Porter 1969 P. Gough 1970 G. Mortimer 1971-1983 Discontinued 1984 H. van der Linde 1985 H. van der Linde 1986 H. van der Linde 1987 J. Coetzee 1988 P. Lanz 1989 J. Coetzee 1990 J. Coetzee 1991 T. Moss 1992 D. Joubert 1993 T. Moss 1994 S. van der Merwe 1995 M. Briggs 1996 T. Moss 1997 G. de Villiers 1998 G. -
An Introduction to Pythagorean Arithmetic
AN INTRODUCTION TO PYTHAGOREAN ARITHMETIC by JASON SCOTT NICHOLSON B.Sc. (Pure Mathematics), University of Calgary, 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming tc^ the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1996 © Jason S. Nicholson, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my i department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Dale //W 39, If96. DE-6 (2/88) Abstract This thesis provides a look at some aspects of Pythagorean Arithmetic. The topic is intro• duced by looking at the historical context in which the Pythagoreans nourished, that is at the arithmetic known to the ancient Egyptians and Babylonians. The view of mathematics that the Pythagoreans held is introduced via a look at the extraordinary life of Pythagoras and a description of the mystical mathematical doctrine that he taught. His disciples, the Pythagore• ans, and their school and history are briefly mentioned. Also, the lives and works of some of the authors of the main sources for Pythagorean arithmetic and thought, namely Euclid and the Neo-Pythagoreans Nicomachus of Gerasa, Theon of Smyrna, and Proclus of Lycia, are looked i at in more detail. -
Group Theory
Group Theory Hartmut Laue Mathematisches Seminar der Universit¨at Kiel 2013 Preface These lecture notes present the contents of my course on Group Theory within the masters programme in Mathematics at the University of Kiel. The aim is to introduce into concepts and techniques of modern group theory which are the prerequisites for tackling current research problems. In an area which has been studied with extreme intensity for many decades, the decision of what to include or not under the time limits of a summer semester was certainly not trivial, and apart from the aspect of importance also that of personal taste had to play a role. Experts will soon discover that among the results proved in this course there are certain theorems which frequently are viewed as too difficult to reach, like Tate’s (4.10) or Roquette’s (5.13). The proofs given here need only a few lines thanks to an approach which seems to have been underestimated although certain rudiments of it have made it into newer textbooks. Instead of making heavy use of cohomological or topological considerations or character theory, we introduce a completely elementary but rather general concept of normalized group action (1.5.4) which serves as a base for not only the above-mentioned highlights but also for other important theorems (3.6, 3.9 (Gasch¨utz), 3.13 (Schur-Zassenhaus)) and for the transfer. Thus we hope to escape the cartesian reservation towards authors in general1, although other parts of the theory clearly follow well-known patterns when a major modification would not result in a gain of clarity or applicability. -
Ideal Theory of the Abelian Group-Algebra
IDEAL THEORY OF THE ABELIAN GROUP-ALGEBRA. ]~Y T. VENK&TARAYUDU, M.A. (University oŸ Madras.) Received September 30, 1937. (Communicated by Dr. R. Vaidyanathaswamy, ~t.A., D.SC.) 1. Introduction. la: is well known 1 that the group-algebra over a commutative corpus K defined by a finite group G is semi-simple, when the characteristic p of the corpus K is not a divisor of the order N of the group and that in this case, the group-algebra K [G] is the direct sum of simple algebras whose products in pairs ate all zero. Beyond these general properties, the structure of the group-algebra is not completely kuown even in the case wheu G is Abelian. For example, in the existing literature, the following questions have not been discussed :-- (1) In the decomposition of K [G] us the direct sum of simple algebras, the actual number of simple component algebras. (2) The radical of K [G] when p is a divisor of N. (3) The residue class ring of K [G] with respect to its radical when it exists. In the present paper, we consider the case when G is Abelian and we show that in this case K [G] is simply isomorphic with the residue class ring of the polynomial domain K [xi, xa, xr] with respect to the ideal (x~'--l, x~~-1, -x~, --1), where ni, na "n~ are the orders of a set of basis elements of the Abelian group G. From a study of this residue class ring, we deduce al1 the important structural properties of the Abelian group-aigebra K [G]. -
Multiplication Fact Strategies Assessment Directions and Analysis
Multiplication Fact Strategies Wichita Public Schools Curriculum and Instructional Design Mathematics Revised 2014 KCCRS version Table of Contents Introduction Page Research Connections (Strategies) 3 Making Meaning for Operations 7 Assessment 9 Tools 13 Doubles 23 Fives 31 Zeroes and Ones 35 Strategy Focus Review 41 Tens 45 Nines 48 Squared Numbers 54 Strategy Focus Review 59 Double and Double Again 64 Double and One More Set 69 Half and Then Double 74 Strategy Focus Review 80 Related Equations (fact families) 82 Practice and Review 92 Wichita Public Schools 2014 2 Research Connections Where Do Fact Strategies Fit In? Adapted from Randall Charles Fact strategies are considered a crucial second phase in a three-phase program for teaching students basic math facts. The first phase is concept learning. Here, the goal is for students to understand the meanings of multiplication and division. In this phase, students focus on actions (i.e. “groups of”, “equal parts”, “building arrays”) that relate to multiplication and division concepts. An important instructional bridge that is often neglected between concept learning and memorization is the second phase, fact strategies. There are two goals in this phase. First, students need to recognize there are clusters of multiplication and division facts that relate in certain ways. Second, students need to understand those relationships. These lessons are designed to assist with the second phase of this process. If you have students that are not ready, you will need to address the first phase of concept learning. The third phase is memorization of the basic facts. Here the goal is for students to master products and quotients so they can recall them efficiently and accurately, and retain them over time.