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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. -
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. -
A Taste of Set Theory for Philosophers
Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. A taste of set theory for philosophers Jouko Va¨an¨ anen¨ ∗ Department of Mathematics and Statistics University of Helsinki and Institute for Logic, Language and Computation University of Amsterdam November 17, 2010 Contents 1 Introduction 1 2 Elementary set theory 2 3 Cardinal and ordinal numbers 3 3.1 Equipollence . 4 3.2 Countable sets . 6 3.3 Ordinals . 7 3.4 Cardinals . 8 4 Axiomatic set theory 9 5 Axiom of Choice 12 6 Independence results 13 7 Some recent work 14 7.1 Descriptive Set Theory . 14 7.2 Non well-founded set theory . 14 7.3 Constructive set theory . 15 8 Historical Remarks and Further Reading 15 ∗Research partially supported by grant 40734 of the Academy of Finland and by the EUROCORES LogICCC LINT programme. I Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. 1 Introduction Originally set theory was a theory of infinity, an attempt to understand infinity in ex- act terms. Later it became a universal language for mathematics and an attempt to give a foundation for all of mathematics, and thereby to all sciences that are based on mathematics. -
The Conjugation Degree on a Set of Metacyclic 3-Groups and 5-Groups with Their Related Graphs
THE CONJUGATION DEGREE ON A SET OF METACYCLIC 3-GROUPS AND 5-GROUPS WITH THEIR RELATED GRAPHS SITI NORZIAHIDAYU AMZEE ZAMRI UNIVERSITI TEKNOLOGI MALAYSIA THE CONJUGATION DEGREE ON A SET OF METACYCLIC 3-GROUPS AND 5-GROUPS WITH THEIR RELATED GRAPHS SITI NORZIAHIDAYU AMZEE ZAMRI A thesis submitted in fulfilment of the requirements for the award of the degree of Doctor of Philosophy Faculty of Science Universiti Teknologi Malaysia SEPTEMBER 2018 iii This thesis is specially dedicated to my dearest family, supervisor, colleagues and friends. iv ACKNOWLEDGEMENT It has been a very tough, memorable yet wonderful experience I had as I was writing and completing my PhD thesis for these three and a half years. For that, I would like to express my deepest gratitude towards my one and only supervisor, Prof Dr Nor Haniza Sarmin, for her unlimited support, patience, guidance and abundance of love as my mother in UTM. Besides, I would also like to thank my co-authors, Dr Sanaa Mohamed Saleh Omer, Dr Mustafa Anis El-Sanfaz and Dr Sanhan Muhammad Salih Khasraw for their help and guidance along my PhD journey. Also, I would like to express my love and thank to my family and UTM friends for their help, care and support. Not to forget my dearest besties, Noraihan Afiqah Rawi and Adnin Afifi Nawi who always had my back and for their never ending love and support. Finally, I would like to thank the Ministry of Higher Education Malaysia for MyPhD scholarship and also to Faculty of Science and Universiti Teknologi Malaysia for the conducive study environment. -
GENERATORS of SPLIT EXTENSIONS of ABELIAN GROUPS by CYCLIC GROUPS3 of Unimodular Rows in M with Size N − 1 Under the Action Of
GENERATORS OF SPLIT EXTENSIONS OF ABELIAN GROUPS BY CYCLIC GROUPS LUC GUYOT Abstract. Let G ≃ M ⋊ C be an n-generator group which is a split ex- tension of an Abelian group M by a cyclic group C. We study the Nielsen equivalence classes and T-systems of generating n-tuples of G. The sub- group M can be turned into a finitely generated faithful module over a suitable quotient R of the integral group ring of C. When C is infinite, we show that the Nielsen equivalence classes of the generating n-tuples of G correspond bijectively to the orbits of unimodular rows in M n−1 under the action of a subgroup of GLn−1(R). Making no assumption on the cardi- nality of C, we exhibit a complete invariant of Nielsen equivalence in the case M ≃ R. As an application, we classify Nielsen equivalence classes and T-systems of soluble Baumslag-Solitar groups, split metacyclic groups and lamplighter groups. 1. Introduction 1.1. Nielsen equivalence related to equivalence of unimodular rows. Given a finitely generated group G, we denote by rk(G) the minimal number of its generators. For n ≥ rk(G), we let Vn(G) be the set of generating n- vectors of G, i.e., the set of elements in Gn whose components generate G. In order to classify generating vectors, we can rely on a well-studied equivalence relation on Vn(G), namely the Nielsen equivalence relation: two generating n-vectors are said to be Nielsen equivalent if they can be related by a finite sequence of transformations taken in the set {Lij,Ii;1 ≤ i =6 j ≤ n} where g arXiv:1604.08896v6 [math.GR] 22 Jun 2018 Lij and Ii replace the component gi of = (g1,...,gn) ∈ Vn(G) by gjgi and −1 gi respectively and leave the other components unchanged. -
On Summability Domains
Proc. Camb. Phil. Soc. (1973), 73, 327 327 PCPS 73-37 Printed in Great Britain On summability domains BY N. J. KALTON Department of Mathematics, University College of Swansea, Singleton Park, Swansea SA2 8PP (Received 12 June 1971) 1. Introduction. We denote by w the space of all complex sequences with the topology given by the semi-norms n where 8 (x) = xn. An FK-space, E, is a subspace of o) on which there exists a complete metrizable locally convex topology T, such that the inclusion (E, T) G O) is con- tinuous; if T is given by a single norm then E is a BK-space. Let A = (a^ii = 1,2, ...;j = 1,2,...) be an infinite matrix; then if a;, yew are such that 00 we write • y = Ax. If E is a sequence space (i.e. a subspace of w) then the summability domain EA is defined to be the set of all a; e o> such that Ax (exists and) belongs to E. If E is an FK- space, with a fundamental sequence of semi-norms (pn), then EA is an FK-space with the topology given^by the semi-norms x ->• q^x) = sup k x-+pn(Ax) (»=1,2,...). We shall denote by 0 the space of all sequences eventually equal to zero. For we write if ^ is an FK-space containing <f> we say that (i) E is a BS-space if (Pnx; n = 1,2,...) is bounded for each xeE, (ii) E is an AK-space if Pnx -> x for each a; G^/. -
A Riesz Decomposition Theorem on Harmonic Spaces Without Positive Potentials
Hiroshima Math. J. 38 (2008), 37–50 A Riesz decomposition theorem on harmonic spaces without positive potentials Ibtesam Bajunaid, Joel M. Cohen, Flavia Colonna and David Singman (Received October 17, 2006) (Revised May 24, 2007) Abstract. In this paper, we give a new definition of the flux of a superharmonic function defined outside a compact set in a Brelot space without positive potentials. We also give a new notion of potential in a BS space (that is, a harmonic space without positive potentials containing the constants) which leads to a Riesz decomposition theorem for the class of superharmonic functions that have a harmonic minorant outside a compact set. Furthermore, we give a characterization of the local axiom of proportionality in terms of a global condition on the space. 1. Introduction The Riesz decomposition theorem for positive superharmonic functions states that any positive superharmonic function on a region of hyperbolic type in C can be expressed uniquely as the sum of a nonnegative potential and a harmonic function. This result is of no interest when the region is parabolic because there are no nonconstant positive superharmonic functions. By a Riesz decomposition theorem for a class of superharmonic functions we mean a unique representation of each member of this class as a global harmonic function plus a member of the class of a special type. For a harmonic space with a Green function, this special class is the class of nonnegative potentials. In this case the most natural class to consider is the admissible superharmonic functions, which are the superharmonic functions that have a harmonic minorant outside a compact set. -
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]. -
Indiscernible Pairs of Countable Sets of Reals at a Given Projective Level
Indiscernible pairs of countable sets of reals at a given projective level Vladimir Kanovei∗ Vassily Lyubetsky† January 1, 2020 Abstract 1 Using an invariant modification of Jensen’s “minimal Π2 singleton” forcing, we define a model of ZFC, in which, for a given n ≥ 2, there exists 1 an Πn unordered pair of non-OD (hence, OD-indiscernible) countable sets 1 of reals, but there is no Σn unordered pairs of this kind. Any two reals x1 6= x2 are discernible by a simple formula ϕ(x) := x < r for a suitable rational r. Therefore, the lowest (type-theoretic) level of sets where one may hope to find indiscernible elements, is the level of sets of reals. And in- deed, identifying the informal notion of definability with the ordinal definability (OD), one finds indiscernible sets of reals in appropriate generic models. Example 1. If reals a 6= b in 2ω form a Cohen-generic pair over L, then ω the constructibility degrees [a]L = {x ∈ 2 : L[x] = L[a]} and [b]L are OD- indiscernible disjoint sets of reals in L[a, b], by rather straightforward forcing arguments, see [2, Theorem 3.1] and a similar argument in [3, Theorem 2.5]. Example 2. As observed in [5], if reals a 6= b in 2ω form a Sacks-generic pair over L, then the constructibility degrees [a]L and [b]L still are OD-indiscernible disjoint sets in L[a, b], with the additional advantage that the unordered pair arXiv:1912.12962v1 [math.LO] 30 Dec 2019 {[a]L, [b]L } is an OD set in L[a, b] because [a]L, [b]L are the only two minimal degrees in L[a, b]. -
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. -
Graduate Prospectus2014 Institute of Space Technology
Graduate Prospectus2014 Institute of Space Technology we HELP YOU ACHIEVE YOUR AMBITIONS P R O S P E C T U S 2 141 INSTITUTE OF SPACE TECHNOLOGY w w w . i s t . e d u . p k To foster intellectual and economic vitality through teaching, research and outreach in the field of Space Science & Technology with a view to improve quality of life. www.ist.edu.pk 2 141 P R O S P E C T U S INSTITUTE OF SPACE TECHNOLOGY CONTENTS Welcome 03 Location 04 Introduction 08 The Institute 09 Facilities 11 Extra Curricular Activities 11 Academic Programs 15 Department of Aeronautics and Astronautics 20 Local MS Programs 22 Linked Programs with Beihang University 31 Linked Programs with Northwestern Polytechnic University 49 Department of Electrical Engineering 51 Local MS Programs 53 Linked Programs with University of Surrey 55 Department of Materials Science & Engineering 72 Department of Mechanical Engineering 81 Department of Remote Sensing & Geo-information Science 100 w w w . i s t . e d u . p k Department of Space Science 106 ORIC 123 Admissions 125 Fee Structure 127 Academic Regulations 130 Faculty 133 Administration 143 Location Map 145 1 20 P R O S P E C T U S 2 141 INSTITUTE OF SPACE TECHNOLOGY w w w . i s t . e d u . p k 2 141 P R O S P E C T U S INSTITUTE OF SPACE TECHNOLOGY Welcome Message Vice Chancellor The Institute of Space Technology welcomes all the students who aspire to enhance their knowledge and specialize in cutting edge technologies. -
The Schur Multiplicator of Metacyclic Groups'
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 40, Number 2, October 1973 THE SCHUR MULTIPLICATOR OF METACYCLIC GROUPS' F. RUDOLF BEYL Abstract. The Schur multiplicator Hfi of a (finite) meta- cyclic group G is computed with the help of the Lyndon spectral sequence. The order of Hfi is a useful invariant of G. A metacyclic group with vanishing Schur multiplicator can be presented with two generators and two relators. A simple description of the totality of these groups is given and all such /»-groups are classified. 1. Introduction. All groups in this note are finite. The group G is called metacyclic if it has a cyclic normal subgroup which yields a cyclic factor group. If M and N are natural numbers, and r and X integers, and r'v=l mod M, then we introduce G(M, N, r, X) = (a, b:aM = 1, b • a ■ZT1 = a\ bN = ß*^^'-i>). Obviously, G(M, N, r, X) is supported by the metacyclic extension (1) e(M, N, r, X):TZMàJL^. g(M, N, r, X) -^-» Cv, where CN is the cyclic group of order N and 2,M the additive cyclic group of order M, and the distinguished generator c of C^ operates on Zw by multiplication by r. (We indicate this by the superscript.) Moreover, 77 maps b to c and a to 1 and k embeds Zjf as the cyclic subgroup of G generated by a. It is well known [6, III §7] that all metacyclic groups can be presented as in (1). Our main result is that the Schur multiplicator H2G(M, N, r, X) is the cyclic group of order (X, h(M, N, r)) where (2) h(M, N, r) = M"1 • (M, r - 1) • (M, 1 + r + • ■■ + r^1) Received by the editors January 29, 1973.