DOCSLIB.ORG

Isologism, Schur-Pair Property and Baer-Invariant of Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Isologism, Schur-Pair Property and Baer-Invariant of Groups World Applied Sciences Journal 16 (11): 1631-1637, 2012 ISSN 1818-4952 © IDOSI Publications, 2012 Isologism, Schur-Pair Property and Baer-Invariant of Groups Roozbeh Modabbernia Islamic Azad University, Shoushtar Branch, Iran Abstract: This paper is devoted to study the connection between the concepts of Schur-pair and the Baer- invariant of groups with respect to a given variety ν of groups. It is shown that if a group G belongs to a certain class of groups then so does its ν-covering group G*. Among other results, a theorem of E.W.Read in 1977 is being generalized to arbitrary varieties of groups. In addition, I consider covering groups and marginal extensions of a ν-perfect group with respect to a subvariety of abelian groups ν and show that any ν-marginal extension of a ν-perfect group G is a homomorphic image of a ν-stem cover of G. Key words: Schur-pair and the baer-invariant of groups • varietal covering groups • subvarieties of abelian groups INTRODUCTION (ii) V( G) = 1iffV*( G) = GiffG ∈ v (iii) [NV*G]= 1iffN ⊆ V*( G) Let F∞ be a free group freely generated by a countable set {x , x ,…}. GGVGN( ) V*GN( ) 1 2 (iv) V= andV*⊇ Let ν be a variety of groups defined by the set of NN N N laws V, which is a subset of F . It will be assumed that V( N) ⊆⊆[ NV*G] N V( G).Inparticular, ∞ (v) the reader is familiar with the notions of verbal V() G=[] GV*G subgroup, V(G) and of marginal subgroup, V*(G), If N V( G) = 1,thenN ⊆ V *( G) associated with a variety of groups ν and agiven group (vi) G. See also [23] for more information on the varieties andV*( G/N )= V* () G / N of groups. (vii) V*( G) V( G) is contained in the Frattini Let G be a group with a normal subgroup N. Then subgroup of G. we define [NV*G] to be the subgroup of G generated by the elements of the following set: The following useful lemma can be proved easily [4]. −1 v( g1i ,...,g n,...,g r)(() v g 1r ,...,g ) Lemma 1.2: Let ν be a variety of groups and G be a group. If G = H N, where H a subgroup and N is a 1≤≤ i r;v ∈ V;g,...,g1r ∈ G;n ∈ N normal subgroup of G, then V (G) = V(H) [NV*G] Let ν be a variety of groups defined by the set of It is easily checked that [NV*G] is the smallest laws V and let G be an arbitrary group with a free normal subgroup T of G contained in N, such that presentation N/T⊆V* (G/T). The following lemma gives basic properties of 1→R→F→G→1 verbal and marginal subgroups of a group G with ν respect to the variety , which are useful in our where F is a free group. Then the Baer-invariant of G investigations [4] for the proofs. with respect to the variety ν, denoted by ν M (G), is defined to be Lemma 1.1: Let ν be a variety of groups defined by the set of laws V and let N be and let N be a nornal R V(F) subgroup of a group G. Then the following statements vM(G)= [RV*F] hold: GG = = One may check that the Baer-invariant of a group (i) V() V *() G 1and V * VG() VG() G is always abelian and independent of the choice of Corresponding Author: Roozbeh Modabbernia, Islamic Azad University, Shoushtar Branch, Iran 1631 World Appl. Sci. J., 16 (11): 1631-1637, 2012 the free presentation of G [7, 8]. In particular, if ν is the Conversely, if V(G) satisfies the descending chain variety of abelian or nilpotent groups of class at most c G condition on normal subgroups and Gv , then (c≥1), then the Baer-invariant of the group G will be N RF ′ N V(G)=<> 1 . (the so called Schur-multiplicator of G) or it []R,F In section 2, we deal with the connection between the Schur pair property and the Baer-invariant of R γc1+ (F) will be , respectively (here γc+1 (F)stands [R , F] groups, In fact, it will be shown that if (ν,x) is a Schur- c * for the (c+1) st term of the lower central series of F pair and G is a ν-covering group of a group G then G∈X if and only if G*∈X (see Corollary 2.3). and [R,c F] = [R,F,…,F], where F is repeated c times [8-10, 12]. In sectin 3, we study the varietal covering groups and among the other results, a theorem of E.W. Read An exact sequence 1→A→G* →G→1 is said to be [24] is being generalized, extensively. Section 4 is a ν-stem cover of G, if (i) A⊆V(G*)∩V*(G*) and (ii) devoted to study the ν-covering groups and ν-marginal A≅νM (G). In this case G* is called a ν-covering group extensions of a ν-perfect group, when ν is taken to be a of G. Note that if ν is taken to be the variety of abelian subvariety of abelian groups. groups, then we have the usual definition of covering group. SCHUR-PAIR AND THE Let ν be a variety of groups defined by the set of BAER-INVARIANT OF GROUPS laws V and let G and H be groups. Then (α,β) is said to be a ν-isologism between G and H, if there exist Let ν be a variety of groups defined by the set of isomorphisms laws V and let X be a class of groups. Then (ν,x) is said to be a Schur-pair, when G is any group with GH α→: G V*(G) V*(H) ∈X it implies that V(G)∈X. V* (G) In particular, if x is the class of all finite groups β → and ; V(G) V(H), such that for all then the above property is known as a Hall’s first conjecture [3]. v(x ,...,x)∈∈ Vandallg,...,g G 1 r 1r In this section we give some equivalent conditions we have that (ν,x) has Schur-pair property if and only if, when β=(v(g,...,g)) v(h,...,h) 1 r 1r G is in x then its Baer-invariant ν M (G) is also in x. For the class of finite groups, we have the whenever hi∈α(giV*(G)), i = 1,…,r. In this case we remarkabletheorem of C.R.Leedham-Green and McKay write G v H and say that G is ν-isologic to H. In [7], which reads as follows : particular, if ν is the variety of abelian groups we obtain the notion of isoclinism due to P. Hall [3, 17, 18]. Theorem 2.1: ([7; Theorem 1.17]) Let ν be a variety of The following lemma of H.N. Hekster [4] is groups defined by the set of laws V and let x be the needed, in our investigation. class of finite groups. Then the following conditions are equivalent: Lemma 1.3: Let ν be a variety of groups defined by the set of laws V and let G be a group with a subgroup H (a) (ν,x) is a Schur-pair ; and a normal subgroup N. (b) For any finite group G, the order of the Baer- Then the following statements hold. invariant of G, |νM(G)|, divides a power of |G|. * * (i) H v HV (G). In particular, if G= HV (G) then Let x be an arbitary class of groups, which is G G v H. Conversely, if satisfies the extension, quotient and normal subgroup closed, i.e. x= V*(G) PQSnX. Then we are able to prove a result similar to descending chain condition on subgroups and Theorem 2.1 for the class x, which is much larger than G v H, then G = HV* (G). the class of finite groups. G G (ii) v . In particular, if N V(G)=<> 1 N V(G) N Theorem 2.2: Let ν be a variety of groups defined by G the set of laws V and let x be a class of groups with x = then G v . N PQSnX. Then the following conditions are equivalent: 1632 World Appl. Sci. J., 16 (11): 1631-1637, 2012 (a) (ν,x) is a Schur-pair; Now, considering this notion we have been able to (b) If G is any group in X, then so is ν M (G). prove a result similar to Theorem 1.3. Proof: Left 1→R→F→G→1 be a free presentation of VARIETAL COVERING GROUPS the group G. Then This section is devoted to study the covering RF groups of a group G, with respect to a given variety of 1→ → →→G1 [RV** F] [ R V F ] groups ν. One should note that, in general, groups might not possess ν-covering groups. In [19], we is a ν-marginal extensionof G. Now if is a Schur-pair presented a class of groups lacking ν-covering groups, and G∈X, then using the property x = Qx and with respect to a certain given variety ν [7]. However, I.Schur in [25] showed the existence of RF covering groups for finite groups and then Jones [26] ⊆ V* [ RV** F ] [RV F] generalized it to every group in the variety of abelian groups.
Load more

Recommended publications
  • Academic Report 2009–10
    Academic Report 2009–10 Harish-Chandra Research Institute Chhatnag Road, Jhunsi, Allahabad 211019 Contents About the Institute 2 Director’s Report 4 Governing Council 8 Academic Staff 10 Administrative Staff 14 Academic Report: Mathematics 16 Academic Report: Physics 47 Workshops and Conferences 150 Recent Graduates 151 Publications 152 Preprints 163 About the Computer Section 173 Library 174 Construction Work 176 1 About the Institute Early Years The Harish-Chandra Research Institute is one of the premier research institute in the country. It is an autonomous institute fully funded by the Department of Atomic Energy, Government of India. Till October 10, 2000 the Institute was known as Mehta Research Institute of Mathematics and Mathematical Physics (MRI) after which it was renamed as Harish-Chandra Research Institute (HRI) after the internationally acclaimed mathematician, late Prof Harish-Chandra. The Institute started with efforts of Dr. B. N. Prasad, a mathematician at the University of Allahabad with initial support from the B. S. Mehta Trust, Kolkata. Dr. Prasad was succeeded in January 1966 by Dr. S. R. Sinha, also of Allahabad University. He was followed by Prof. P. L. Bhatnagar as the first formal Director. After an interim period in January 1983, Prof. S. S. Shrikhande joined as the next Director of the Institute. During his tenure the dialogue with the Department of Atomic Energy (DAE) entered into decisive stage and a review committee was constituted by the DAE to examine the Institutes fu- ture. In 1985 N. D. Tiwari, the then Chief Minister of Uttar Pradesh, agreed to provide sufficient land for the Institute and the DAE promised financial sup- port for meeting both the recurring and non-recurring expenditure.
    [Show full text]
  • Varieties of Groups and Isologisms
    J. Austral. Math. Soc. (Series A) 46 (1989), 22-60 VARIETIES OF GROUPS AND ISOLOGISMS N. S. HEKSTER (Received 2 February 1987) Communicated by H. Lausch Abstract In order to classify solvable groups Philip Hall introduced in 1939 the concept of isoclinism. Subsequently he defined a more general notion called isologism. This is so to speak isoclinism with respect to a certain variety of groups. The equivalence relation isologism partitions the class of all groups into families. The present paper is concerned with the internal structure of these families. 1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision): 20 E 10, 20 E 99, 20 F 14. Keywords and phrases: variety of groups, isologism. 1. Introduction, preliminaries and notational conventions It is a well-known fact that solvable groups are difficult to classify because of the abundance of normal subgroups. In a certain sense the class of groups of prime- power order is the simplest case to handle. But already here the classification is far from being complete. Up to now there are only a few classes of such prime- power order groups which have been adequately analysed. The first to create some order in the plethora of groups of prime-power order was Philip Hall. He observed that the notion of isomorphism of groups is really too strong to give rise to a satisfactory classification and that it had to be replaced by a weaker equivalence relation. Subsequently he discovered a suitable equivalence relation and called it isoclinism of groups (see [6]). It is this classification principle that 1989 Australian Mathematical Society 0263-6115/89 SA2.00 + 0.00 22 Downloaded from https://www.cambridge.org/core.
    [Show full text]
  • Italian Journal of Pure and Applied Mathematics ISSN 2239-0227
    N° 28 – July 2011 Italian Journal of Pure and Applied Mathematics ISSN 2239-0227 EDITOR-IN-CHIEF Piergiulio Corsini Editorial Board Saeid Abbasbandy Antonino Giambruno Ivo Rosenberg Reza Ameri Furio Honsell Paolo Salmon Luisa Arlotti James Jantosciak Maria Scafati Tallini Krassimir Atanassov Tomas Kepka Kar Ping Shum Malvina Baica David Kinderlehrer Alessandro Silva Federico Bartolozzi Andrzej Lasota Sergio Spagnolo Rajabali Borzooei Violeta Leoreanu Hari M. Srivastava Carlo Cecchini Mario Marchi Marzio Strassoldo Gui-Yun Chen Donatella Marini Yves Sureau Domenico Nico Chillemi Angelo Marzollo Carlo Tasso Stephen Comer Antonio Maturo Ioan Tofan Irina Cristea Jean Mittas Aldo Ventre Mohammad Reza Darafsheh M. Reza Moghadam Thomas Vougiouklis Bal Kishan Dass Petr Nemec Hans Weber Bijan Davvaz Vasile Oproiu Yunqiang Yin Alberto Felice De Toni Livio C. Piccinini Mohammad Mehdi Zahedi Franco Eugeni Goffredo Pieroni Fabio Zanolin Giovanni Falcone Flavio Pressacco Paolo Zellini Yuming Feng Vito Roberto Jianming Zhan F O R U M EDITOR-IN-CHIEF Franco Eugeni Vito Roberto Dipartimento di Metodi Quantitativi per l'Economia del Territorio Dipartimento di Matematica e Informatica Piergiulio Corsini Università di Teramo, Italy Via delle Scienze 206 - 33100 Udine, Italy [email protected] [email protected] Department of Civil Engineering and Architecture Via delle Scienze 206 - 33100 Udine, Italy Giovanni Falcone Ivo Rosenberg [email protected] Dipartimento di Metodi e Modelli Matematici Departement de Mathematique et de Statistique viale delle Scienze Ed. 8 Université de Montreal 90128 Palermo, Italy C.P. 6128 Succursale Centre-Ville VICE-CHIEF [email protected] Montreal, Quebec H3C 3J7 - Canada [email protected] Yuming Feng Violeta Leoreanu College of Math.
    [Show full text]
  • Isoclinism in Probability of Commuting N-Tuples 1 Introduction
    Isoclinism in Probability of Commuting n-tuples A. Erfanian and F. Russo Department of Mathematics, Centre of Excellency in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, Mashhad, Iran. [email protected] Department of Mathematics, University of Naples, Naples, Italy. [email protected] Abstract Strong restrictions on the structure of a group G can be given, once that it is known the probability that a randomly chosen pair of elements of a finite group G commutes. Introducing the notion of mutually commuting n-tuples for compact groups (not necessary finite), the present paper generalizes the probability that a randomly chosen pair of elements of G commutes. We shall state some results concerning this new concept of probability which has been recently treated in [3]. Furthermore a relation has been found between the notion of mutually commuting n-tuples and that of isoclinism between two arbitrary groups. Mathematics Subject Classification: Primary: 20D60, 20P05; Sec- ondary: 20D08. Keywords: Mutually commuting pairs,commuting n-tuples, commutativ- ity degree, isoclinic groups. 1 Introduction Let G be a finite group, then the probability that a randomly chosen pair of elements of G commutes is defined to be #com(G)=jGj2, where #com(G) is 2 A. Erfanian and F. Russo the number of pairs (x; y) 2 G × G = G2 with xy = yx and will be briefly denoted by cp(G). From [6], one may easily find that cp(G) = k(G)=jGj, where k(G) is the number of conjugacy classes of G. So, there is no ambiguity to use one or the other ratio in the universe of finite groups.
    [Show full text]
  • AUTOMORPHISMS of GROUPS December, 2014
    AUTOMORPHISMS OF GROUPS By PRADEEP KUMAR RAI MATH08200904006 Harish-Chandra Research Institute, Allahabad A thesis submitted to the Board of Studies in Mathematical Sciences In partial fulfillment of requirements for the Degree of DOCTOR OF PHILOSOPHY of HOMI BHABHA NATIONAL INSTITUTE December, 2014 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfilment of requirements for an advanced degree at Homi Bhabha National Institute (HBNI) and is deposited in the Library to be made available to borrowers under rules of the HBNI. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgement of source is made. Requests for permis- sion for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the Competent Authority of HBNI when in his or her judgement the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. Pradeep Kumar Rai DECLARATION I, hereby declare that the investigation presented in the thesis has been carried out by me. The work is original and has not been submitted earlier as a whole or in part for a degree / diploma at this or any other Institution / University. Pradeep Kumar Rai List of Publications arising from the thesis Journal 1.“On finite p-groups with abelian automorphism group”, Vivek K. Jain, Pradeep K. Rai and Manoj K. Yadav, Internat. J. Algebra Comput., 2014, Vol. 23, 1063-1077. 2.“On class-preserving automorphisms of groups”, Pradeep K. Rai, Ricerche Mat., 2014, Vol. 63, 189-194.
    [Show full text]
  • Isoclinism and Weak Isoclinism Invariants 1. Introduction Isoclinism
    Isoclinism and weak isoclinism invariants STEPHEN M. BUCKLEY Abstract. We find new family invariants for group isoclinism, and also explore weak isoclinism. 1. Introduction Isoclinism is an equivalence relation for groups that was introduced by Hall [4], and is widely used in the group theory literature. The isoclinism equiva- lence class containing a given group G is called the family of G. There are many known family invariants, i.e. properties possessed by all members of a family F if they are possessed by any one member of F. The isomorphism type of the commutator subgroup G0 of a group G is a family invariant, but that of the center Z(G) is not. Lescot [6, Lemma 2.8] showed that the order of G0\Z(G) is a family invariant. Here we strengthen that result by showing that the isomorphism type of G0 \ Z(G) is a family invariant. 0 Theorem 1.1. If the groups G1 and G2 are isoclinic, then G1 \ Z(G1) and 0 G2 \ Z(G2) are isomorphic. Let 1 ≤ iAb(G) ≤ 1 be the minimal index of an abelian subgroup in a group G. Desmond MacHale conjectured that iAb(G) is a family invariant, and we show that this is indeed true. Proposition 1.2. If the groups G1 and G2 are isoclinic, then iAb(G1) = iAb(G2). After some preliminaries in Section 2, we prove the above results in Section 3. In Section 4, we compare isoclinism with what we call weak isoclinism. 2. Preliminaries Throughout the remainder of the paper, G is always a group.
    [Show full text]
  • Finite Rings with Many Commuting Pairs of Elements
    Finite rings with many commuting pairs of elements STEPHEN M. BUCKLEY, DESMOND MACHALE, AND AINE´ N´I SHE´ Abstract. We investigate the set of values attained by Pr(R), the probability that a random pair of elements in a finite ring R commute. A key tool is a new notion of isoclinism for rings, and an associated canonical form for rings. In particular, we show that Pr(R) is an isoclinic invariant, and characterize all possible values of Pr(R) ≥ 11=32, and the associated isoclinism families. 1. Introduction There has been much written on the possible values attained by the probability that a random pair of elements in a finite group commute: see for instance [5], [11], [7], [14], [17], [13], [4], [6], [3], and [9]. In this paper, we examine the corresponding question for finite rings. This topic, by contrast, has attracted little attention: indeed [15] is the only contribution of which we are aware. To be precise, we define the commuting probability to be jf(x; y) 2 R × R : xy = yxgj Pr(R) := jRj2 where R is a finite ring, and jSj denotes cardinality of a set S. Let R be the set of values of Pr(R) as R ranges over all (possibly non-unital) finite rings. Trivially, R ⊂ (0; 1] \ Q. For each prime p, we define Rp similarly, except that R is allowed to range only over p-rings, meaning rings whose order is a power of p. Throughout the paper, we write p2 + p − 1 2p2 − 1 p3 + p2 − 1 α = ; β = ; and γ = : p p3 p p4 p p5 We will see later that 1 γ < α2 < β < < α ; p ≥ 2 : p p p p p We determine in particular all t 2 R such that t ≥ γ2 = 11=32, and all t 2 Rp, t ≥ γp, where p is prime.
    [Show full text]
  • Propagation of Artin Patterns Between Isoclinic 2-Groups
    PROPAGATION OF ARTIN TRANSFER PATTERNS BETWEEN ISOCLINIC 2-GROUPS DANIEL C. MAYER Abstract. For isoclinism families Φ of finite non-abelian 2-groups, it is investigated by which general laws the abelian type invariants and the transfer kernels of the stem Φ(0) are connected with those of the branches Φ(n) with n ≥ 1. 1. Introduction As opposed to groups with odd prime power orders, finite 2-groups G with abelianizations G=G0 of the types (2; 2), (4; 2) and (2; 2; 2) possess sufficiently simple presentations to admit an easy calculation of invariants like central series, two-step centralizers, maximal subgroups, Artin trans- fers, and Artin patterns. Isoclinism between stem groups of type (2; 2) and branch groups of the types (4; 2) and (2; 2; 2) ensures isomorphic central quotients and isomorphic lower central series of the former and latter [12], and is useful for comparing Artin patterns [19, Dfn. 4.3, p. 27]. 2. Isoclinism of groups For classifying finite p-groups, Hall introduced the concept of isoclinism [12] which admits to collect non-isomorphic groups with certain similarities in isoclinism families. Hall's idea was based on the following connection between commutators and central quotients (inner automorphisms). Lemma 2.1. Let G be a group, then the commutator mapping [:; :]: G × G ! G0, (g; h) 7! [g; h], factorizes through the central quotient by inducing a well-defined map 0 (2.1) cG :(G/ζG) × (G/ζG) ! G ; (g · ζG; h · ζG) 7! [g; h]: Proof. Let ! : G ! G/ζG be the natural projection onto the central quotient.
    [Show full text]
  • Crossed Modules and Cat1-Groups 2.82
    XMod Crossed Modules and Cat1-Groups 2.82 23 October 2020 Chris Wensley Murat Alp Alper Odabas Enver Onder Uslu XMod 2 Chris Wensley Email: [email protected] Homepage: http://pages.bangor.ac.uk/~mas023/ Address: Dr. C.D. Wensley School of Computer Science and Electronic Engineering Bangor University Dean Street Bangor Gwynedd LL57 1UT UK Murat Alp Email: [email protected] Address: Prof. Dr. M. Alp Ömer Halisdemir University Art and Science Faculty Mathematics Department Nigde Turkey Alper Odabas Email: [email protected] Address: Dr. A. Odabas Osmangazi University Arts and Sciences Faculty Department of Mathematics and Computer Science Eskisehir Turkey XMod 2 Abstract The XMod package provides functions for computation with • finite crossed modules of groups and cat1-groups, and morphisms of these structures; • finite pre-crossed modules, pre-cat1-groups, and their Peiffer quotients; • isoclinism classes of groups and crossed modules; • derivations of crossed modules and sections of cat1-groups; • crossed squares and their morphisms, including the actor crossed square of a crossed module; • crossed modules of finite groupoids (experimental version). XMod was originally implemented in 1996 using the GAP3 language, when the second author was studying for a Ph.D. [Alp97] in Bangor. In April 2002 the first and third parts were converted to GAP4, the pre-structures were added, and version 2.001 was released. The final two parts, covering derivations, sections and actors, were included in the January 2004 release 2.002 for GAP 4.4. In October 2015 functions for computing isoclinism classes of crossed modules, written by Alper Odaba¸s and Enver Uslu, were added.
    [Show full text]
  • The University of Chicago Lazard Correspondence Up
    THE UNIVERSITY OF CHICAGO LAZARD CORRESPONDENCE UP TO ISOCLINISM A DISSERTATION SUBMITTED TO THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS BY VIPUL NAIK CHICAGO, ILLINOIS DECEMBER 2013 TABLE OF CONTENTS LIST OF TABLES . xii ABSTRACT . xiii ACKNOWLEDGMENTS . xiv 1 INTRODUCTION, OUTLINE, AND PRELIMINARIES . 1 1.0.1 Background assumed . 1 1.0.2 Group and subgroup notation . 1 1.0.3 Lie ring and subring notation . 3 1.0.4 Other conventions . 4 1.1 Introduction . 4 1.1.1 The difference in tractability between groups and abelian groups . 4 1.1.2 Nilpotent groups and their relation with abelian groups . 5 1.1.3 The Lie correspondence: general remarks . 7 1.1.4 The Lie algebra for the general linear group . 8 1.1.5 The nilpotent case of the Lie correspondence . 9 1.1.6 The example of the unitriangular matrix group . 10 1.1.7 The Malcev correspondence and Lazard correspondence . 11 1.1.8 Our generalization of the Lazard correspondence . 12 1.1.9 Similarities and differences between groups and Lie rings . 12 1.1.10 Our central tool: Schur multipliers . 13 1.1.11 Globally and locally nilpotent . 14 1.1.12 The structure of this document . 15 1.1.13 For a quick reading . 17 1.2 Outline of our main results . 21 1.2.1 The Lazard correspondence: a rapid review . 21 1.2.2 Isoclinism: a rapid review . 22 1.2.3 The Lazard correspondence up to isoclinism .
    [Show full text]
  • Extended Abstracts Utm Kuala Lumpur, Malaysia | 23-26 January 2017
    PROCEEDINGS OF THE FOURTH BIENNIAL INTERNATIONAL GROUP THEORY CONFERENCE 2017 (4BIGTC2017) EXTENDED ABSTRACTS UTM KUALA LUMPUR, MALAYSIA | 23-26 JANUARY 2017 Organized by Applied Algebra and Analysis Research Group (AAAG) in collaboration with Faculty of Science & UTM International Proceedings of The Fourth Biennial International Group Theory Conference 2017 (4BIGTC2017) 23-26 January 2017 Extended Abstracts Universiti Teknologi Malaysia Malaysia Preface The Biennial International Group Theory Conference (BIGTC) is a series of conference that is organized every two years, and has been hosted by three countries - Malaysia, Turkey and Iran. The aim of this conference is to exchange the research ideas amongst the group theorists and postgraduate students. In addition, specialists in the subject of group theory can present their latest research works and encourage interested students to progress and widen their knowledge in group theory. Certainly this provides great opportunities for the graduate students, junior researchers and specialist in the region, which end up with joint research works internationally. The First, Second and Third BIGTC have already taken place in Malaysia (UTM Johor Bahru), Turkey (Istanbul) and Iran (Mashhad), respectively. Once again, this Fourth Biennial International Group Theory Conference 2017 (4BIGTC2017) is hosting by Universiti Teknologi Malaysia (UTM) but this time in UTM Kuala Lumpur. We do hope that others countries besides Malaysia, Turkey and Iran will also join us in organizing BIGTC in the future. This book contains collection of extended abstracts in the broad subject of group theory. All extended abstracts, which will be presented in the plenary, invited or parallel talks in the conference, have been collected in this booklet for the convenience of the participants.
    [Show full text]
  • On Finite Groups All of Whose Irreducible Complex Characters Are Primitive
    MATHEMATICS Proceedings A 88 (1), March 25, 1985 On finite groups all of whose irreducible complex characters are primitive by N.S. Hekster Math. Institute, Univ. of Amsterdam, Roetersstraat 15, 1018 WB Amsterdam, the Netherlands Communicated by Prof. H.W. Lenstra Jr. at the meeting of October 29, 1984 ABSTRACT A finite group is called a PC-group if none of its irreducible complex characters is induced by any character of a proper subgroup. The structure of PC-groups is determined. Hereby the classification of the finite simple groups plays an essential role. 1. INTRODUCTION In studying the complex character theory of a finite group G one will notice that some of the irreducible characters of G can be obtained by inducing a character of a proper subgroup, whereas others cannot be obtained in this way. Hereby two extreme cases occur. On the one hand it can happen that every irreducible character of G is induced from a linear (i.e. degree 1) character of a subgroup. The group G is then called an M-group, as all its irreducible characters are monomial. On the other hand it can happen that none of the irreducible characters of G can be obtained by inducing any character of a proper subgroup. Let us call such a group G a PC-group, for all its irreducible characters are primitive. The main object of our paper will be the determination of the structure of PC-groups. It turns out that although the definition of PC-groups is character theoretic, a purely group theoretic description of PC-groups can be given.
    [Show full text]