DOCSLIB.ORG

Finite Simple Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Finite Simple Groups Finite Simple Groups Daniel Rogers Why do we care about simple groups? What do we know about simple groups? Finite Simple Groups What questions are there about groups in light of the classifcation? The extension Daniel Rogers problem Maximal subgroups October 28 2014 Contents Finite Simple Groups Daniel Rogers Why do we care about simple groups? 1 Why do we care about simple groups? What do we know about simple groups? What questions 2 are there about What do we know about simple groups? groups in light of the classifcation? The extension problem Maximal 3 What questions are there about groups in light of the subgroups classifcation? The extension problem Maximal subgroups Important note! Finite Simple Groups Daniel Rogers Why do we care about simple groups? What do we know about simple groups? What questions are there about All groups in this talk will be finite. groups in light of the classifcation? The extension problem Maximal subgroups Prime Numbers Finite Simple Groups Daniel Rogers Why do we care about simple groups? What do we know about simple groups? Prime numbers are the building blocks of number theory - every What questions integer can be expressed uniquely as a product of primes. As such, a are there about groups in light of lot of effort goes in to understanding prime numbers. the classifcation? The extension Simple groups are the equivalent notion in group theory - although, problem Maximal as we will see later, there are some crucial differences. subgroups Theorem The only too simple groups are cyclic groups of prime order (and the trivial group). This definition is, as the name suggests, too simple. First attempt Finite Simple Groups Daniel Rogers Why do we care about simple groups? What do we Definition know about simple groups? A group G is too simple if the only subgroups of G are G and 1. What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups This definition is, as the name suggests, too simple. First attempt Finite Simple Groups Daniel Rogers Why do we care about simple groups? What do we Definition know about simple groups? A group G is too simple if the only subgroups of G are G and 1. What questions are there about groups in light of the classifcation? Theorem The extension problem The only too simple groups are cyclic groups of prime order (and the Maximal subgroups trivial group). First attempt Finite Simple Groups Daniel Rogers Why do we care about simple groups? What do we Definition know about simple groups? A group G is too simple if the only subgroups of G are G and 1. What questions are there about groups in light of the classifcation? Theorem The extension problem The only too simple groups are cyclic groups of prime order (and the Maximal subgroups trivial group). This definition is, as the name suggests, too simple. Key definitions Finite Simple Groups Daniel Rogers Why do we care Definition about simple groups? A normal subgroup N of a group G is a subgroup of G which is What do we know about closed under conjugation by elements of G; in other words, simple groups? 8 n 2 N; g 2 G, g −1ng 2 N. What questions are there about groups in light of the classifcation? Definition The extension problem Maximal A group G is simple if it has precisely two normal subgroup; namely subgroups G and 1. Example For p prime, the cyclic group of order p (denoted Cp) is simple. Jordan-H¨oldertheorem Finite Simple Groups Daniel Rogers Why do we care Definition about simple groups? A composition series for a finite group G is a chain of strict subgroups What do we know about simple groups? 1 = G0 < G1 < ::: < Gr = G What questions are there about groups in light of such that each Gi is a normal subgroup of Gi+1 and the factor group the classifcation? The extension Gi+1=Gi is simple. problem Maximal subgroups Definition The composition factors for a group is the set of groups fG1=G0; G2=G1; :::; Gr =Gr−1g Every (finite) group has a composition series. ±SL(2; 5), the group of all matrices with determinant ±1 (order 240). SL(2; 5), the group of all matrices with determinant 1 (order 120). Z, the group of all scalar matrices (order 4). 1 2 Z, the group consisting of I , the identity, and −I (order 2). 1, the trivial group (order 1). Then we have the following two composition series: 1 1 E 2 Z E Z E ±SL(2; 5) E G 1 1 E 2 Z E SL(2; 5) E ±SL(2; 5) E G. Jordan-H¨oldertheorem Finite Simple Groups Example (GL(2,5)) Daniel Rogers Let G = GL(2; 5), the set of invertible 2 × 2 matrices over the Why do we care about simple integers modulo 5 (this is a group of order 480). G has the following groups? What do we normal subgroups: know about simple groups? What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups Then we have the following two composition series: 1 1 E 2 Z E Z E ±SL(2; 5) E G 1 1 E 2 Z E SL(2; 5) E ±SL(2; 5) E G. Jordan-H¨oldertheorem Finite Simple Groups Example (GL(2,5)) Daniel Rogers Let G = GL(2; 5), the set of invertible 2 × 2 matrices over the Why do we care about simple integers modulo 5 (this is a group of order 480). G has the following groups? What do we normal subgroups: know about simple groups? ±SL(2; 5), the group of all matrices with determinant ±1 (order What questions 240). are there about groups in light of the classifcation? SL(2; 5), the group of all matrices with determinant 1 (order The extension problem 120). Maximal subgroups Z, the group of all scalar matrices (order 4). 1 2 Z, the group consisting of I , the identity, and −I (order 2). 1, the trivial group (order 1). Jordan-H¨oldertheorem Finite Simple Groups Example (GL(2,5)) Daniel Rogers Let G = GL(2; 5), the set of invertible 2 × 2 matrices over the Why do we care about simple integers modulo 5 (this is a group of order 480). G has the following groups? What do we normal subgroups: know about simple groups? ±SL(2; 5), the group of all matrices with determinant ±1 (order What questions 240). are there about groups in light of the classifcation? SL(2; 5), the group of all matrices with determinant 1 (order The extension problem 120). Maximal subgroups Z, the group of all scalar matrices (order 4). 1 2 Z, the group consisting of I , the identity, and −I (order 2). 1, the trivial group (order 1). Then we have the following two composition series: 1 1 E 2 Z E Z E ±SL(2; 5) E G 1 1 E 2 Z E SL(2; 5) E ±SL(2; 5) E G. Jordan-H¨oldertheorem Finite Simple Groups Daniel Rogers Why do we care about simple groups? What do we know about Example (GL(2,5)) simple groups? What questions 1 are there about 1 E 2 Z E Z E ±SL(2; 5) E G groups in light of 1 the classifcation? 1 E 2 Z E SL(2; 5) E ±SL(2; 5) E G. The extension problem Maximal These composition series, although different, have the same (multiset subgroups ∼ of) composition factors, namely fC2; C2; C2; PSL(2; 5) = Alt(5)g. Jordan-H¨oldertheorem Finite Simple Groups Daniel Rogers Why do we care about simple groups? What do we Theorem (Jordan-H¨older) know about simple groups? Any two composition series for a group G have the same composition What questions are there about factors, up to permutation and isomorphism. groups in light of the classifcation? The extension This gives us a well-defined notion of 'factors' of a group, somewhat problem Maximal equivalent to the notion of primes in number theory. Thus, by subgroups understanding all simple groups we understand all the factors of a group. The Classification of Finite Simple Groups (CFSG) Finite Simple Groups Daniel Rogers Why do we care about simple groups? What do we know about simple groups? What questions are there about The classification of finite simple groups is a question which took groups in light of the classifcation? over a century from proposal to proof. The extension problem Maximal subgroups 1870 - Camille Jordan discovers 4 classes of simple groups, which we now call the classical groups, over fields of prime order. 1873 - Emile´ Mathieu discovers 5 'sporadic' simple groups (ones that are not part of infinite families). 1892 - Otto H¨olderfirst asks for a classification of finite simple groups. CFSG history Finite Simple Groups Daniel Rogers 1832 - Evariste´ Galois defines Why do we care a normal subgroup and proves about simple groups? that Alt(n) is simple for What do we n > 4. know about simple groups? What questions are there about groups in light of the classifcation? The extension problem Maximal subgroups 1873 - Emile´ Mathieu discovers 5 'sporadic' simple groups (ones that are not part of infinite families). 1892 - Otto H¨olderfirst asks for a classification of finite simple groups. CFSG history Finite Simple Groups Daniel Rogers 1832 - Evariste´ Galois defines Why do we care a normal subgroup and proves about simple groups? that Alt(n) is simple for What do we n > 4. know about simple groups? 1870 - Camille Jordan What questions are there about discovers 4 classes of simple groups in light of the classifcation? groups, which we now call the The extension problem classical groups, over fields of Maximal subgroups prime order.
Load more

Recommended publications
  • Classification of Finite Abelian Groups
    Math 317 C1 John Sullivan Spring 2003 Classification of Finite Abelian Groups (Notes based on an article by Navarro in the Amer. Math. Monthly, February 2003.) The fundamental theorem of finite abelian groups expresses any such group as a product of cyclic groups: Theorem. Suppose G is a finite abelian group. Then G is (in a unique way) a direct product of cyclic groups of order pk with p prime. Our first step will be a special case of Cauchy’s Theorem, which we will prove later for arbitrary groups: whenever p |G| then G has an element of order p. Theorem (Cauchy). If G is a finite group, and p |G| is a prime, then G has an element of order p (or, equivalently, a subgroup of order p). ∼ Proof when G is abelian. First note that if |G| is prime, then G = Zp and we are done. In general, we work by induction. If G has no nontrivial proper subgroups, it must be a prime cyclic group, the case we’ve already handled. So we can suppose there is a nontrivial subgroup H smaller than G. Either p |H| or p |G/H|. In the first case, by induction, H has an element of order p which is also order p in G so we’re done. In the second case, if ∼ g + H has order p in G/H then |g + H| |g|, so hgi = Zkp for some k, and then kg ∈ G has order p. Note that we write our abelian groups additively. Definition. Given a prime p, a p-group is a group in which every element has order pk for some k.
    [Show full text]
  • Mathematics 310 Examination 1 Answers 1. (10 Points) Let G Be A
    Mathematics 310 Examination 1 Answers 1. (10 points) Let G be a group, and let x be an element of G. Finish the following definition: The order of x is ... Answer: . the smallest positive integer n so that xn = e. 2. (10 points) State Lagrange’s Theorem. Answer: If G is a finite group, and H is a subgroup of G, then o(H)|o(G). 3. (10 points) Let ( a 0! ) H = : a, b ∈ Z, ab 6= 0 . 0 b Is H a group with the binary operation of matrix multiplication? Be sure to explain your answer fully. 2 0! 1/2 0 ! Answer: This is not a group. The inverse of the matrix is , which is not 0 2 0 1/2 in H. 4. (20 points) Suppose that G1 and G2 are groups, and φ : G1 → G2 is a homomorphism. (a) Recall that we defined φ(G1) = {φ(g1): g1 ∈ G1}. Show that φ(G1) is a subgroup of G2. −1 (b) Suppose that H2 is a subgroup of G2. Recall that we defined φ (H2) = {g1 ∈ G1 : −1 φ(g1) ∈ H2}. Prove that φ (H2) is a subgroup of G1. Answer:(a) Pick x, y ∈ φ(G1). Then we can write x = φ(a) and y = φ(b), with a, b ∈ G1. Because G1 is closed under the group operation, we know that ab ∈ G1. Because φ is a homomorphism, we know that xy = φ(a)φ(b) = φ(ab), and therefore xy ∈ φ(G1). That shows that φ(G1) is closed under the group operation.
    [Show full text]
  • Math 412. Simple Groups
    Math 412. Simple Groups DEFINITION: A group G is simple if its only normal subgroups are feg and G. Simple groups are rare among all groups in the same way that prime numbers are rare among all integers. The smallest non-abelian group is A5, which has order 60. THEOREM 8.25: A abelian group is simple if and only if it is finite of prime order. THEOREM: The Alternating Groups An where n ≥ 5 are simple. The simple groups are the building blocks of all groups, in a sense similar to how all integers are built from the prime numbers. One of the greatest mathematical achievements of the Twentieth Century was a classification of all the finite simple groups. These are recorded in the Atlas of Simple Groups. The mathematician who discovered the last-to-be-discovered finite simple group is right here in our own department: Professor Bob Greiss. This simple group is called the monster group because its order is so big—approximately 8 × 1053. Because we have classified all the finite simple groups, and we know how to put them together to form arbitrary groups, we essentially understand the structure of every finite group. It is difficult, in general, to tell whether a given group G is simple or not. Just like determining whether a given (large) integer is prime, there is an algorithm to check but it may take an unreasonable amount of time to run. A. WARM UP. Find proper non-trivial normal subgroups of the following groups: Z, Z35, GL5(Q), S17, D100.
    [Show full text]
  • Chapter 25 Finite Simple Groups
    Chapter 25 Finite Simple Groups Chapter 25 Finite Simple Groups Historical Background Definition A group is simple if it has no nontrivial proper normal subgroup. The definition was proposed by Galois; he showed that An is simple for n ≥ 5 in 1831. It is an important step in showing that one cannot express the solutions of a quintic equation in radicals. If possible, one would factor a group G as G0 = G, find a normal subgroup G1 of maximum order to form G0/G1. Then find a maximal normal subgroup G2 of G1 and get G1/G2, and so on until we get the composition factors: G0/G1,G1/G2,...,Gn−1/Gn, with Gn = {e}. Jordan and Hölder proved that these factors are independent of the choices of the normal subgroups in the process. Jordan in 1870 found four infinite series including: Zp for a prime p, SL(n, Zp)/Z(SL(n, Zp)) except when (n, p) = (2, 2) or (2, 3). Between 1982-1905, Dickson found more infinite series; Miller and Cole showed that 5 (sporadic) groups constructed by Mathieu in 1861 are simple. Chapter 25 Finite Simple Groups In 1950s, more infinite families were found, and the classification project began. Brauer observed that the centralizer has an order 2 element is important; Feit-Thompson in 1960 confirmed the 1900 conjecture that non-Abelian simple group must have even order. From 1966-75, 19 new sporadic groups were found. Thompson developed many techniques in the N-group paper. Gorenstein presented an outline for the classification project in a lecture series at University of Chicago in 1972.
    [Show full text]
  • The Theory of Finite Groups: an Introduction (Universitext)
    Universitext Editorial Board (North America): S. Axler F.W. Gehring K.A. Ribet Springer New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo This page intentionally left blank Hans Kurzweil Bernd Stellmacher The Theory of Finite Groups An Introduction Hans Kurzweil Bernd Stellmacher Institute of Mathematics Mathematiches Seminar Kiel University of Erlangen-Nuremburg Christian-Albrechts-Universität 1 Bismarckstrasse 1 /2 Ludewig-Meyn Strasse 4 Erlangen 91054 Kiel D-24098 Germany Germany [email protected] [email protected] Editorial Board (North America): S. Axler F.W. Gehring Mathematics Department Mathematics Department San Francisco State University East Hall San Francisco, CA 94132 University of Michigan USA Ann Arbor, MI 48109-1109 [email protected] USA [email protected] K.A. Ribet Mathematics Department University of California, Berkeley Berkeley, CA 94720-3840 USA [email protected] Mathematics Subject Classification (2000): 20-01, 20DXX Library of Congress Cataloging-in-Publication Data Kurzweil, Hans, 1942– The theory of finite groups: an introduction / Hans Kurzweil, Bernd Stellmacher. p. cm. — (Universitext) Includes bibliographical references and index. ISBN 0-387-40510-0 (alk. paper) 1. Finite groups. I. Stellmacher, B. (Bernd) II. Title. QA177.K87 2004 512´.2—dc21 2003054313 ISBN 0-387-40510-0 Printed on acid-free paper. © 2004 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis.
    [Show full text]
  • Quasi P Or Not Quasi P? That Is the Question
    Rose-Hulman Undergraduate Mathematics Journal Volume 3 Issue 2 Article 2 Quasi p or not Quasi p? That is the Question Ben Harwood Northern Kentucky University, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Harwood, Ben (2002) "Quasi p or not Quasi p? That is the Question," Rose-Hulman Undergraduate Mathematics Journal: Vol. 3 : Iss. 2 , Article 2. Available at: https://scholar.rose-hulman.edu/rhumj/vol3/iss2/2 Quasi p- or not quasi p-? That is the Question.* By Ben Harwood Department of Mathematics and Computer Science Northern Kentucky University Highland Heights, KY 41099 e-mail: [email protected] Section Zero: Introduction The question might not be as profound as Shakespeare’s, but nevertheless, it is interesting. Because few people seem to be aware of quasi p-groups, we will begin with a bit of history and a definition; and then we will determine for each group of order less than 24 (and a few others) whether the group is a quasi p-group for some prime p or not. This paper is a prequel to [Hwd]. In [Hwd] we prove that (Z3 £Z3)oZ2 and Z5 o Z4 are quasi 2-groups. Those proofs now form a portion of Proposition (12.1) It should also be noted that [Hwd] may also be found in this journal. Section One: Why should we be interested in quasi p-groups? In a 1957 paper titled Coverings of algebraic curves [Abh2], Abhyankar conjectured that the algebraic fundamental group of the affine line over an algebraically closed field k of prime characteristic p is the set of quasi p-groups, where by the algebraic fundamental group of the affine line he meant the family of all Galois groups Gal(L=k(X)) as L varies over all finite normal extensions of k(X) the function field of the affine line such that no point of the line is ramified in L, and where by a quasi p-group he meant a finite group that is generated by all of its p-Sylow subgroups.
    [Show full text]
  • A STUDY on the ALGEBRAIC STRUCTURE of SL 2(Zpz)
    A STUDY ON THE ALGEBRAIC STRUCTURE OF SL2 Z pZ ( ~ ) A Thesis Presented to The Honors Tutorial College Ohio University In Partial Fulfillment of the Requirements for Graduation from the Honors Tutorial College with the degree of Bachelor of Science in Mathematics by Evan North April 2015 Contents 1 Introduction 1 2 Background 5 2.1 Group Theory . 5 2.2 Linear Algebra . 14 2.3 Matrix Group SL2 R Over a Ring . 22 ( ) 3 Conjugacy Classes of Matrix Groups 26 3.1 Order of the Matrix Groups . 26 3.2 Conjugacy Classes of GL2 Fp ....................... 28 3.2.1 Linear Case . .( . .) . 29 3.2.2 First Quadratic Case . 29 3.2.3 Second Quadratic Case . 30 3.2.4 Third Quadratic Case . 31 3.2.5 Classes in SL2 Fp ......................... 33 3.3 Splitting of Classes of(SL)2 Fp ....................... 35 3.4 Results of SL2 Fp ..............................( ) 40 ( ) 2 4 Toward Lifting to SL2 Z p Z 41 4.1 Reduction mod p ...............................( ~ ) 42 4.2 Exploring the Kernel . 43 i 4.3 Generalizing to SL2 Z p Z ........................ 46 ( ~ ) 5 Closing Remarks 48 5.1 Future Work . 48 5.2 Conclusion . 48 1 Introduction Symmetries are one of the most widely-known examples of pure mathematics. Symmetry is when an object can be rotated, flipped, or otherwise transformed in such a way that its appearance remains the same. Basic geometric figures should create familiar examples, take for instance the triangle. Figure 1: The symmetries of a triangle: 3 reflections, 2 rotations. The red lines represent the reflection symmetries, where the trianlge is flipped over, while the arrows represent the rotational symmetry of the triangle.
    [Show full text]
  • 17 Lagrange's Theorem
    Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 17 Lagrange's Theorem A very important corollary to the fact that the left cosets of a subgroup partition a group is Lagrange's Theorem. This theorem gives a relationship between the order of a finite group G and the order of any subgroup of G(in particular, if jGj < 1 and H ⊆ G is a subgroup, then jHj j jGj). Theorem 17.1 (Lagrange's Theorem) Let G be a finite group, and let H be a subgroup of G: Then the order of H divides the order of G: Proof. By Theorem 16.1, the right cosets of H form a partition of G: Thus, each element of G belongs to at least one right coset of H in G; and no element can belong to two distinct right cosets of H in G: Therefore every element of G belongs to exactly one right coset of H. Moreover, each right coset of H contains jHj elements (Lemma 16.2). Therefore, jGj = njHj; where n is the number of right cosets of H in G: Hence, jHj j jGj: This ends a proof of the theorem. Example 17.1 If jGj = 14 then the only possible orders for a subgroup are 1, 2, 7, and 14. Definition 17.1 The number of different right cosets of H in G is called the index of H in G and is denoted by [G : H]: It follows from the above definition and the proof of Lagrange's theorem that jGj = [G : H]jHj: Example 17.2 6 Since jS3j = 3! = 6 and j(12)j = j < (12) > j = 2 then [S3; < (12) >] = 2 = 3: 1 The rest of this section is devoted to consequences of Lagrange's theorem; we begin with the order of an element.
    [Show full text]
  • Order (Group Theory) 1 Order (Group Theory)
    Order (group theory) 1 Order (group theory) In group theory, a branch of mathematics, the term order is used in two closely-related senses: • The order of a group is its cardinality, i.e., the number of its elements. • The order, sometimes period, of an element a of a group is the smallest positive integer m such that am = e (where e denotes the identity element of the group, and am denotes the product of m copies of a). If no such m exists, a is said to have infinite order. All elements of finite groups have finite order. The order of a group G is denoted by ord(G) or |G| and the order of an element a by ord(a) or |a|. Example Example. The symmetric group S has the following multiplication table. 3 • e s t u v w e e s t u v w s s e v w t u t t u e s w v u u t w v e s v v w s e u t w w v u t s e This group has six elements, so ord(S ) = 6. By definition, the order of the identity, e, is 1. Each of s, t, and w 3 squares to e, so these group elements have order 2. Completing the enumeration, both u and v have order 3, for u2 = v and u3 = vu = e, and v2 = u and v3 = uv = e. Order and structure The order of a group and that of an element tend to speak about the structure of the group.
    [Show full text]
  • The Mathieu Groups (Simple Sporadic Symmetries)
    The Mathieu Groups (Simple Sporadic Symmetries) Scott Harper (University of St Andrews) Tomorrow's Mathematicians Today 21st February 2015 Scott Harper The Mathieu Groups 21st February 2015 1 / 15 The Mathieu Groups (Simple Sporadic Symmetries) Scott Harper (University of St Andrews) Tomorrow's Mathematicians Today 21st February 2015 Scott Harper The Mathieu Groups 21st February 2015 2 / 15 1 2 A symmetry is a structure preserving permutation of the underlying set. A group acts faithfully on an object if it is isomorphic to a subgroup of the 4 3 symmetry group of the object. Symmetry group: D4 The stabiliser of a point in a group G is Group of rotations: the subgroup of G which fixes x. ∼ h(1 2 3 4)i = C4 Subgroup fixing 1: h(2 4)i Symmetry Scott Harper The Mathieu Groups 21st February 2015 3 / 15 A symmetry is a structure preserving permutation of the underlying set. A group acts faithfully on an object if it is isomorphic to a subgroup of the symmetry group of the object. The stabiliser of a point in a group G is Group of rotations: the subgroup of G which fixes x. ∼ h(1 2 3 4)i = C4 Subgroup fixing 1: h(2 4)i Symmetry 1 2 4 3 Symmetry group: D4 Scott Harper The Mathieu Groups 21st February 2015 3 / 15 A group acts faithfully on an object if it is isomorphic to a subgroup of the symmetry group of the object. The stabiliser of a point in a group G is Group of rotations: the subgroup of G which fixes x.
    [Show full text]
  • Finite Abelian P-Primary Groups
    CLASSIFICATION OF FINITE ABELIAN GROUPS 1. The main theorem Theorem 1.1. Let A be a finite abelian group. There is a sequence of prime numbers p p p 1 2 ··· n (not necessarily all distinct) and a sequence of positive integers a1,a2,...,an such that A is isomorphic to the direct product ⇠ a1 a2 an A Zp Zp Zpn . ! 1 ⇥ 2 ⇥···⇥ In particular n A = pai . | | i iY=n Example 1.2. We can classify abelian groups of order 144 = 24 32. Here are the possibilities, with the partitions of the powers of 2 and 3 on⇥ the right: Z2 Z2 Z2 Z2 Z3 Z3;(4, 2) = (1 + 1 + 1 + 1, 1 + 1) ⇥ ⇥ ⇥ ⇥ ⇥ Z2 Z2 Z4 Z3 Z3;(4, 2) = (1 + 1 + 2, 1 + 1) ⇥ ⇥ ⇥ ⇥ Z4 Z4 Z3 Z3;(4, 2) = (2 + 2, 1 + 1) ⇥ ⇥ ⇥ Z2 Z8 Z3 Z3;(4, 2) = (1 + 3, 1 + 1) ⇥ ⇥ ⇥ Z16 Z3 Z3;(4, 2) = (4, 1 + 1) ⇥ ⇥ Z2 Z2 Z2 Z2 Z9;(4, 2) = (1 + 1 + 1 + 1, 2) ⇥ ⇥ ⇥ ⇥ Z2 Z2 Z4 Z9;(4, 2) = (1 + 1 + 2, 2) ⇥ ⇥ ⇥ Z4 Z4 Z9;(4, 2) = (2 + 2, 2) ⇥ ⇥ Z2 Z8 Z9;(4, 2) = (1 + 3, 2) ⇥ ⇥ Z16 Z9 cyclic, isomorphic to Z144;(4, 2) = (4, 2). ⇥ There are 10 non-isomorphic abelian groups of order 144. Theorem 1.1 can be broken down into two theorems. 1 2CLASSIFICATIONOFFINITEABELIANGROUPS Theorem 1.3. Let A be a finite abelian group. Let q1,...,qr be the distinct primes dividing A ,andsay | | A = qbj . | | j Yj Then there are subgroups A A, j =1,...,r,with A = qbj ,andan j ✓ | j| j isomorphism A ⇠ A A A .
    [Show full text]
  • Linear Algebraic Groups
    Clay Mathematics Proceedings Volume 4, 2005 Linear Algebraic Groups Fiona Murnaghan Abstract. We give a summary, without proofs, of basic properties of linear algebraic groups, with particular emphasis on reductive algebraic groups. 1. Algebraic groups Let K be an algebraically closed field. An algebraic K-group G is an algebraic variety over K, and a group, such that the maps µ : G × G → G, µ(x, y) = xy, and ι : G → G, ι(x)= x−1, are morphisms of algebraic varieties. For convenience, in these notes, we will fix K and refer to an algebraic K-group as an algebraic group. If the variety G is affine, that is, G is an algebraic set (a Zariski-closed set) in Kn for some natural number n, we say that G is a linear algebraic group. If G and G′ are algebraic groups, a map ϕ : G → G′ is a homomorphism of algebraic groups if ϕ is a morphism of varieties and a group homomorphism. Similarly, ϕ is an isomorphism of algebraic groups if ϕ is an isomorphism of varieties and a group isomorphism. A closed subgroup of an algebraic group is an algebraic group. If H is a closed subgroup of a linear algebraic group G, then G/H can be made into a quasi- projective variety (a variety which is a locally closed subset of some projective space). If H is normal in G, then G/H (with the usual group structure) is a linear algebraic group. Let ϕ : G → G′ be a homomorphism of algebraic groups. Then the kernel of ϕ is a closed subgroup of G and the image of ϕ is a closed subgroup of G.
    [Show full text]