An Introduction to Finite Simple Groups
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On the Classification of Finite Simple Groups
On the Classification of Finite Simple Groups Niclas Bernhoff Division for Engineering Sciences, Physics and Mathemathics Karlstad University April6,2004 Abstract This paper is a small note on the classification of finite simple groups for the course ”Symmetries, Groups and Algebras” given at the Depart- ment of Physics at Karlstad University in the Spring 2004. 1Introduction In 1892 Otto Hölder began an article in Mathematische Annalen with the sen- tence ”It would be of the greatest interest if it were possible to give an overview of the entire collection of finite simple groups” (translation from German). This can be seen as the starting point of the classification of simple finite groups. In 1980 the following classification theorem could be claimed. Theorem 1 (The Classification theorem) Let G be a finite simple group. Then G is either (a) a cyclic group of prime order; (b) an alternating group of degree n 5; (c) a finite simple group of Lie type;≥ or (d) one of 26 sporadic finite simple groups. However, the work comprises in total about 10000-15000 pages in around 500 journal articles by some 100 authors. This opens for the question of possible mistakes. In fact, around 1989 Aschbacher noticed that an 800-page manuscript on quasithin groups by Mason was incomplete in various ways; especially it lacked a treatment of certain ”small” cases. Together with S. Smith, Aschbacher is still working to finish this ”last” part of the proof. Solomon predicted this to be finished in 2001-2002, but still until today the paper is not completely finished (see the homepage of S. -
Construction of Finite Matrix Groups
Construction of finite matrix groups Robert A. Wilson School of Mathematics and Statistics, The University of Birmingham published in Birkh¨auserProgress in Math. 173 (1999), pp. 61{83 Abstract We describe various methods of construction of matrix representa- tions of finite groups. The applications are mainly, but not exclusively, to quasisimple or almost simple groups. Some of the techniques can also be generalized to permutation representations. 1 Introduction It is one thing to determine the characters of a group, but quite another to construct the associated representations. For example, it is an elementary exercise to obtain the character table of the alternating group A5 by first determining the conjugacy classes, then writing down the trivial character and the permutation characters on points and unordered pairs, and using row orthogonality to obtain the irreducibles of degree 4 and 5, and finally using column orthogonality to complete the table. The result (see Table 1) shows that there are two characters of degree 3, but how do we construct the corresponding 3-dimensional representations? In general, we need some more information than just the characters, such as a presentation in terms of generators and relations, or some knowledge of subgroup structure, such as a generating amalgam, or something similar. If we have a presentation for our group, then in a sense it is already de- termined, and there are various algorithms which in principle at least will construct more or less any desired representation. The most important and 1 Table 1: The character table of A5 Class name 1A 2A 3A 5A 5B Class size 1 15 20 12 12 χ1 1 1 1 1 1 χ2 3 −1 0 τ σ χ3 3 −1 0 σ τ χ4 4 0 1 −1 −1 χ5 5 1 −1 0 0 1 p 1 p τ = (1 + 5); σ = (1 − 5) 2 2 well-known is Todd{Coxeter coset enumeration, which converts a presenta- tion into a permutation representation, and is well described in many places, such as [11]. -
Janko's Sporadic Simple Groups
Janko’s Sporadic Simple Groups: a bit of history Algebra, Geometry and Computation CARMA, Workshop on Mathematics and Computation Terry Gagen and Don Taylor The University of Sydney 20 June 2015 Fifty years ago: the discovery In January 1965, a surprising announcement was communicated to the international mathematical community. Zvonimir Janko, working as a Research Fellow at the Institute of Advanced Study within the Australian National University had constructed a new sporadic simple group. Before 1965 only five sporadic simple groups were known. They had been discovered almost exactly one hundred years prior (1861 and 1873) by Émile Mathieu but the proof of their simplicity was only obtained in 1900 by G. A. Miller. Finite simple groups: earliest examples É The cyclic groups Zp of prime order and the alternating groups Alt(n) of even permutations of n 5 items were the earliest simple groups to be studied (Gauss,≥ Euler, Abel, etc.) É Evariste Galois knew about PSL(2,p) and wrote about them in his letter to Chevalier in 1832 on the night before the duel. É Camille Jordan (Traité des substitutions et des équations algébriques,1870) wrote about linear groups defined over finite fields of prime order and determined their composition factors. The ‘groupes abéliens’ of Jordan are now called symplectic groups and his ‘groupes hypoabéliens’ are orthogonal groups in characteristic 2. É Émile Mathieu introduced the five groups M11, M12, M22, M23 and M24 in 1861 and 1873. The classical groups, G2 and E6 É In his PhD thesis Leonard Eugene Dickson extended Jordan’s work to linear groups over all finite fields and included the unitary groups. -
ELEMENTARY ABELIAN P-SUBGROUPS of ALGEBRAIC GROUPS
ROBERT L. GRIESS, JR ELEMENTARY ABELIAN p-SUBGROUPS OF ALGEBRAIC GROUPS Dedicated to Jacques Tits for his sixtieth birthday ABSTRACT. Let ~ be an algebraically closed field and let G be a finite-dimensional algebraic group over N which is nearly simple, i.e. the connected component of the identity G O is perfect, C6(G °) = Z(G °) and G°/Z(G °) is simple. We classify maximal elementary abelian p-subgroups of G which consist of semisimple elements, i.e. for all primes p ~ char K. Call a group quasisimple if it is perfect and is simple modulo the center. Call a subset of an algebraic group total if it is in a toms; otherwise nontoral. For several quasisimple algebraic groups and p = 2, we define complexity, and give local criteria for whether an elementary abelian 2-subgroup of G is total. For all primes, we analyze the nontoral examples, include a classification of all the maximal elementary abelian p-groups, many of the nonmaximal ones, discuss their normalizers and fusion (i.e. how conjugacy classes of the ambient algebraic group meet the subgroup). For some cases, we give a very detailed discussion, e.g. p = 3 and G of type E6, E 7 and E 8. We explain how the presence of spin up and spin down elements influences the structure of projectively elementary abelian 2-groups in Spin(2n, C). Examples of an elementary abelian group which is nontoral in one algebraic group but toral in a larger one are noted. Two subsets of a maximal torus are conjugate in G iff they are conjugate in the normalizer of the torus; this observation, with our discussion of the nontoral cases, gives a detailed guide to the possibilities for the embedding of an elementary abelian p-group in G. -
Completing the Brauer Trees for the Sporadic Simple Lyons Group
COMPLETING THE BRAUER TREES FOR THE SPORADIC SIMPLE LYONS GROUP JURGEN¨ MULLER,¨ MAX NEUNHOFFER,¨ FRANK ROHR,¨ AND ROBERT WILSON Abstract. In this paper we complete the Brauer trees for the sporadic sim- ple Lyons group Ly in characteristics 37 and 67. The results are obtained using tools from computational representation theory, in particular a new con- densation technique, and with the assistance of the computer algebra systems MeatAxe and GAP. 1. Introduction 1.1. In this paper we complete the Brauer trees for the sporadic simple Lyons group Ly in characteristics 37 and 67. The results are stated in Section 2, and will also be made accessible in the character table library of the computer algebra system GAP and electronically in [1]. The shape of the Brauer trees as well as the labelling of nodes up to algebraic conjugacy of irreducible ordinary characters had already been found in [8, Section 6.19.], while here we complete the trees by determining the labelling of the nodes on their real stems and their planar embedding; proofs are given in Section 4. Together with the results in [8, Section 6.19.] for the other primes dividing the group order, this completes all the Brauer trees for Ly. Our main computational workhorse is fixed point condensation, which originally was invented for permutation modules in [20], but has been applied to different types of modules as well. To our knowledge, the permutation module we have condensed is the largest one for which this has been accomplished so far. The theoretical background of the idea of condensation is described in Section 3. -
Biased Tadpoles: a Fast Algorithm for Centralizers in Large Matrix Groups
Biased Tadpoles: a Fast Algorithm for Centralizers in Large Matrix Groups Daniel Kunkle and Gene Cooperman College of Computer and Information Science Northeastern University Boston, MA, USA [email protected], [email protected] ABSTRACT of basis formula in linear algebra. Given a group G, the def Centralizers are an important tool in in computational group centralizer subgroup of g G is defined as G(g) = h: h h ∈ C { ∈ theory. Yet for large matrix groups, they tend to be slow. G, g = g . } We demonstrate a O(p G (1/ log ε)) black box randomized Note that the above definition of centralizer is well-defined algorithm that produces| a| centralizer using space logarith- independently of the representation of G. Even for permu- mic in the order of the centralizer, even for typical matrix tation group representations, it is not known whether the groups of order 1020. An optimized version of this algorithm centralizer problem can be solved in polynomial time. Nev- (larger space and no longer black box) typically runs in sec- ertheless, in the permutation group case, Leon produced a onds for groups of order 1015 and minutes for groups of or- partition backtrack algorithm [8, 9] that is highly efficient der 1020. Further, the algorithm trivially parallelizes, and so in practice both for centralizer problems and certain other linear speedup is achieved in an experiment on a computer problems also not known to be in polynomial time. Theißen with four CPU cores. The novelty lies in the use of a bi- produced a particularly efficient implementation of partition ased tadpole, which delivers an order of magnitude speedup backtrack for GAP [17]. -
Arxiv:1305.5974V1 [Math-Ph]
INTRODUCTION TO SPORADIC GROUPS for physicists Luis J. Boya∗ Departamento de F´ısica Te´orica Universidad de Zaragoza E-50009 Zaragoza, SPAIN MSC: 20D08, 20D05, 11F22 PACS numbers: 02.20.a, 02.20.Bb, 11.24.Yb Key words: Finite simple groups, sporadic groups, the Monster group. Juan SANCHO GUIMERA´ In Memoriam Abstract We describe the collection of finite simple groups, with a view on physical applications. We recall first the prime cyclic groups Zp, and the alternating groups Altn>4. After a quick revision of finite fields Fq, q = pf , with p prime, we consider the 16 families of finite simple groups of Lie type. There are also 26 extra “sporadic” groups, which gather in three interconnected “generations” (with 5+7+8 groups) plus the Pariah groups (6). We point out a couple of physical applications, in- cluding constructing the biggest sporadic group, the “Monster” group, with close to 1054 elements from arguments of physics, and also the relation of some Mathieu groups with compactification in string and M-theory. ∗[email protected] arXiv:1305.5974v1 [math-ph] 25 May 2013 1 Contents 1 Introduction 3 1.1 Generaldescriptionofthework . 3 1.2 Initialmathematics............................ 7 2 Generalities about groups 14 2.1 Elementarynotions............................ 14 2.2 Theframeworkorbox .......................... 16 2.3 Subgroups................................. 18 2.4 Morphisms ................................ 22 2.5 Extensions................................. 23 2.6 Familiesoffinitegroups ......................... 24 2.7 Abeliangroups .............................. 27 2.8 Symmetricgroup ............................. 28 3 More advanced group theory 30 3.1 Groupsoperationginspaces. 30 3.2 Representations.............................. 32 3.3 Characters.Fourierseries . 35 3.4 Homological algebra and extension theory . 37 3.5 Groupsuptoorder16.......................... -
Construction of the Rudvalis Group of Order 145,926,144,000*
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 27, 538-548 (1973) Construction of the Rudvalis Group of Order 145,926,144,000* J. H. CONWAY Department of Pure Mathematics and Mathematical Statistics, Cambridge, England AND D. B. WALES Department of Mathematics, California Institute of Technology, Pasadena, California 91109 Communicated by Marshall Hall, Jr. Received August 4, 1972 Recently, Arunas Rudvalis [l] provided evidence for the existence of a new simple group R of order 145,926,144,000 = 2r4 * 33 * 5a . 7 * 13 . 29. He describes the group as a rank 3 permutation group on 4060 letters, in which the stabilizer of a point is the (nonsimple) Ree group F = V’,(2), which has orbits of sizes 1,1755,2304, corresponding to subgroups of F of orders 20480 and 15600. The first of these is the centralizer of an involution of F, while the second has the known subgroup L = PSLa(25) of P’ (see Ref. [4]) as a subgroup of index 2. Since one of the involutions in R has no fixed point and so just 2030 2-cycles, an argument of Griess and Schur [2] shows that R has a proper double cover 2R. Rudvalis and Frame gave evidence for supposing that this group had a 28-dimensional complex representation not splitting over F. (Feit and Lyons have proved this under further assumptions on R.) We sup- pose that this unitary representation of 2R exists while constructing the group, but then conclude independently of this supposition that the con- struction defines a group 2R whose central quotient is R. -
New Criteria for the Solvability of Finite Groups
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 77. 234-246 (1982) New Criteria for the Solvability of Finite Groups ZVI ARAD DePartment of Malhematics, Bar-llan Utzioersity. Kamat-Gun. Israel AND MICHAEL B. WARD Department of Mathemalics, Bucknell University, Lewisburg, Penrtsylcania I7837 Communicared by Waher Feif Received September 27. 1981 1. INTRODUCTION All groups considered in this paper are finite. Suppose 7~is a set of primes. A subgroup H of a group G is called a Hall 7r-subgroup of G if every prime dividing ] HI is an element of x and no prime dividing [G : H ] is an element of 71.When p is a prime, p’ denotes the set of all primes different from p. In Hall’s famous characterization of solvable groups he proves that a group is solvable if and only if it has a Hall p’-subgroup for each prime p (see [6] or [ 7, p. 291]). It has been conjectured that a group is solvable if and only if it has a Hall 2’-subgroup and a Hall 3’subgroup (see [ 11). We verify this conjecture for groups whose composition factors are known simple groups. Along the way we also verify (for groups whose composition factors are known simple groups) a conjecture of Hall’s that if a group has a Hall ( p, q}-subgroup for every pair of primes p, 9, then the group is solvable [7, p. 2911. Thus, if one assumes the classification of finite simple groups, then both of these conjectures are settled in the affirmative. -
Finite Simple Groups Which Projectively Embed in an Exceptional Lie Group Are Classified!
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 36, Number 1, January 1999, Pages 75{93 S 0273-0979(99)00771-5 FINITE SIMPLE GROUPS WHICH PROJECTIVELY EMBED IN AN EXCEPTIONAL LIE GROUP ARE CLASSIFIED! ROBERT L. GRIESS JR. AND A. J. E. RYBA Abstract. Since finite simple groups are the building blocks of finite groups, it is natural to ask about their occurrence “in nature”. In this article, we consider their occurrence in algebraic groups and moreover discuss the general theory of finite subgroups of algebraic groups. 0. Introduction Group character theory classifies embeddings of finite groups into classical groups. No general theory classifies embeddings into the exceptional complex al- gebraic group, i.e., one of G2(C), F4(C), E6(C), E7(C), E8(C). For exceptional groups, special methods seem necessary. Since the early 80s, there have been ef- forts to determine which central extensions of finite simple groups embed in an exceptional group. For short, we call this work a study of projective embeddings of finite simple groups into exceptional groups. Table PE on page 84 contains a summary. The classification program for finite subgroups of complex algebraic groups involves both existence of embeddings and their classification up to conjugacy. We have just classified embeddings of Sz(8) into E8(C) (there are three, up to E8(C)- conjugacy), thus settling the existence question for projective embeddings of finite simple groups into exceptional algebraic groups. The conjugacy part of the program is only partially resolved. The finite subgroups of the smallest simple algebraic group PSL(2; C)(upto conjugacy) constitute the famous list: cyclic, dihedral, Alt4, Sym4, Alt5. -
Arxiv:1804.05648V2 [Math.GR] 8 Aug 2018 Xml,Ta Si H Prdcsml Ru Fhl.Terlvn Sub Relevant the 104]
A NEGATIVE ANSWER TO A QUESTION OF ASCHBACHER ROBERT A. WILSON Abstract. We give infinitely many examples to show that, even for simple groups G, it is possible for the lattice of overgroups of a subgroup H to be the Boolean lattice of rank 2, in such a way that the two maximal overgroups of H are conjugate in G. This answers negatively a question posed by Aschbacher. 1. The question In a recent survey article on the subgroup structure of finite groups [1], in the context of discussing open problems on the possible structures of subgroup lattices of finite groups, Aschbacher poses the following specific question. Let G be a finite group, H a subgroup of G, and suppose that H is contained in exactly two maximal subgroups M1 and M2 of G, and that H is maximal in both M1 and M2. Does it follow that M1 and M2 are not conjugate in G? This is Question 8.1 in [1]. For G a general group, he asserts there is a counterexample, not given in [1], so he restricts this question to the case G almost simple, that is S ≤ G ≤ Aut(S) for some simple group S. This is Question 8.2 in [1]. 2. The answer In fact, the answer is no, even for simple groups G. The smallest example seems to be the simple Mathieu group M12 of order 95040. Theorem 1. Let G = M12, and H =∼ A5 acting transitively on the 12 points permuted by M12. Then H lies in exactly two other subgroups of G, both lying in the single conjugacy class of maximal subgroups L2(11). -
Moonshines for L2(11) and M12
Moonshines for L2(11) and M12 Suresh Govindarajan Department of Physics Indian Institute of Technology Madras (Work done with my student Sutapa Samanta) Talk at the Workshop on Modular Forms and Black Holes @NISER, Bhubaneswar on January 13. 2017 Plan Introduction Some finite group theory Moonshine BKM Lie superalgebras Introduction Classification of Finite Simple Groups Every finite simple group is isomorphic to one of the following groups: (Source: Wikipedia) I A cyclic group with prime order; I An alternating group of degree at least 5; I A simple group of Lie type, including both I the classical Lie groups, namely the groups of projective special linear, unitary, symplectic, or orthogonal transformations over a finite field; I the exceptional and twisted groups of Lie type (including the Tits group which is not strictly a group of Lie type). I The 26 sporadic simple groups. The classification was completed in 2004 when Aschbacher and Smith filled the last gap (`the quasi-thin case') in the proof. Fun Reading: Symmetry and the Monster by Mark Ronan The sporadic simple groups I the Mathieu groups: M11, M12, M22, M23, M24; (found in 1861) I the Janko groups: J1, J2, J3, J4; (others 1965-1980) I the Conway groups; Co1, Co2, Co3; 0 I the Fischer groups; Fi22. Fi23, Fi24; I the Higman-Sims group; HS I the McLaughlin group: McL I the Held group: He; I the Rudvalis group Ru; I the Suzuki sporadic group: Suz; 0 I the O'Nan group: O N; I Harada-Norton group: HN; I the Lyons group: Ly; I the Thompson group: Th; I the baby Monster group: B and Sources: Wikipedia and Mark Ronan I the Fischer-Griess Monster group: M Monstrous Moonshine Conjectures I The j-function has the followed q-series: (q = exp(2πiτ)) j(τ)−744 = q−1+[196883+1] q+[21296876+196883+1] q2+··· I McKay observed that 196883 and 21296876 are the dimensions of the two smallest irreps of the Monster group.