DOCSLIB.ORG

Characterization of Δ-Small Submodule

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Characterization of Δ-Small Submodule Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.5, No.7, 2015 Characterization of δ-small submodule R. S. Wadbude, M.R.Aloney & Shubhanka Tiwari 1.Mahatma Fule Arts, Commerce and Sitaramji Chaudhari Science Mahavidyalaya, Warud. SGB Amaravati University Amravati [M.S.] 2.Technocrats Institute of Technology (excellence) Barkttulla University Bhopal. [M.P.] Abstract Pseudo Projectivity and M- Pseudo Projectivity is a generalization of Projevtevity. [2], [8] studied M-Pseudo Projective module and small M-Pseudo Projective module. In this paper we consider some generalization of small M-Pseudo Projective module, that is δ-small M-Pseudo Projective module with the help of δ-small and δ- cover. Key words: Singular module, S.F. small M-Pseudo Projective module, δ-small and projective δ- cover. Introduction: Throughout this paper R is an associative ring with unity module and all modules are unitary left R- modules. A sub module K of a module M. K ≤ M. Let M be a module, K ≤ M is said to be small in M if for every L ≤ M, the equality K + L = M implies L = M, ( denoted by 퐾 ≪ 푀 ). The concept of δ-small sub modules was introduced by Zhon [10]. A sub module K of M is said to be δ-small sub module of M (denoted by 퐾 ≪훿 푀 ) if whenever M = K+ L, with M/K is singular, then M = L. The sum of all δ-small sub module of M is denoted by δ(M). δ(M) is the reject in M of class of all singular simple modules. 훿(푀) = 푅푒푗푀(℘) =∩ 푀 {푁 ≤ 푀: ∈ ℘}, where ℘ be the class of all singular modules. (푅푒푗 (ℚ) is the intersection of all K ≤ M, with 푁 푀 M/K torsion free). An R-module M is said to be hollow (δ-hollow) if all proper sub modules of M are small (δ- small) in M. An R-module M is S.F. if zero is only small sub module in M. G. Azumay introduced projective cover. W.xue [12] generalized projective cover. A module epimorphism f : P M is a cover in case ker f Rad(P), 퐾푒푟푓 ≪ 푀. A cover f : P M is called a protective cover in case P is projective module. An epimorphism f : P M is called a δ-projective cover of module M in case 퐾푒푟푓 ≪훿 훿(푃) and P is projective. A δ-cover f : P M of a module M, is said to be a self projective δ-cover in case p is self projective module. Projective cover is denoted by P(M), if there is an epimorphism with P(M) is projective and 푘푒푟푓 ≪ 푃(푀). fM : P(M) M 푀 In last section we introducing a new characterization of small M-pseudo projective module. We prove that N is hollow, then N is δ-small M-pseudo projective module if and only if N is M-pseudo projective module, and let M be a δ-small pseudo projective module then M is S.F. if and only if M/A is isomorphic to direct summand of M, A ≤ M. 1.δ-Small Definition:1.1. The sub module Z(M) {xM : rR (x)is essential in R}is called singular sub module of M. The module M is Called a singular module if Z(M) = M. The module M is Called a non- singular module if Z(M) = 0. Definition:1.2. An R-module N is called small M- pseudo projective module if for every sub module A of M, M any epimorphism f : N with 퐾푒푟푓 ≪ 푁, Can be lifted to a homomorphism h: N M A N h f M M 0 g A 0 i.e. g h f . 18 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.5, No.7, 2015 Definition:1.3 An R-module N is called δ-small M- pseudo projective module if for every sub module A of M, any small epimorphism with 퐾푒푟푓 ≪훿 푁, Can be lifted to a homomorphism N h f epic with krf in N M M 0 g smallepic. A 0 i.e. g h f . Examples:1 i) Every small sub module of M is δ-small in M. ii) Every non singular semisimple sub module of M is δ-small in M. iii) Every simple module M is hollow. iv) Z6 as Z-module is not δ-small. Example:2. Consider the Z-modules Z4 and Z2 . An epimorphism f : Z4 Z2 define by f (1) f (3) 1 and f (0) f (2) 0 Then f is small epimorphism . Every small sub module of M is δ-small in M. Example:3. Let R M Z6 . Then two non-trivial sub modules of M, M1 0,3and M 2 0,2,4 are δ-small in M, but neither M 1 and M 2 is small in M, Moreover 푀 ≪훿 푀 Proposition:1.1 Let M, L and N be R-Modules. If : M N and : N L are two epimorphisms. Then is small if and only if both α, β are small. Proof: [8] Lemma:1.1. Let N be a submodule of M. The following are equivalent: i) 푁 ≪훿 푀 ii) If X + N = M, then M = X Y for a projective semi simple sub module Y with Y N. iii) If X + N = M, with M/X Goldie torsion, then X = M. Proof: [10]. Lemma:1.2. Let M be a module, then i) For sub module N, K, L with K ≤ N, We have a) 푁 ≪훿 푀 if and only if 퐾 ≪훿 푀 and 푁/퐾 ≪훿 푀/퐾 b) 푁 + 퐿 ≪훿 푀 if and only if 푁 ≪훿 푀 and 퐿 ≪훿 푀. ii) If 퐾 ≪훿 푀 and f : M N is an homomorphism, then 푓(퐾) ≪훿 푁, In particular, if 퐾 ≪훿 푀 ≤ 푁 and 퐾 ≪훿 푁. iii) Let K1 M1 M, K2 M2 M and M M1 M 2 , then 퐾1 ⊕ 퐾2 ≪훿 푀1 ⊕ 푀1 푖푓 푎푛푑 표푛푙푦 푖푓 퐾1 ≪훿 푀1 푎푛푑 퐾2 ≪훿 푀2. Proof: [4]. Lemma:1.3. Let M be a module. Then 푀 i) 훿(푀) = ∑{퐿 ≤ 푀: 퐿 ≪ 푀} =∩ {퐾 ≤ 푀 ∶ 푖푠 푠푖푛푔푢푙푎푟 푚표푑푢푙푒} . M 훿 퐾 ii) If f: M→Nf is: Nan R-homomorphism, then 푓(훿(푀)) ≤ 훿(푁). Therefore δ(M) ish :fullyN M invariant sub moduleA of M. In particular if K ≤ M, then δ(K) ≤ δ(M). iii) 퐼푓 푀 =⊕푖∈퐼 (푀푖), 푡ℎ푒푛 훿(푀) ≤⊕푖∈퐼 훿(푀푖). iv) If every proper sub module of M is contained in a maximal sub module of m, then δ(M) is the unique largest δ-small sub module of M. In particular if M is finitely generated, then δ(M) is δ- small in M. Proof: [4] Lemma:1.4. If K ≤ N ≤ M, 퐾 ≪훿 푀 and N is a direct summand of M, Then 퐾 ≪훿 푁. Proof: [10] Proposition: Given a module M, each of the following sets is equal δ(M). i) 훿(푀) = ∑{퐴: 퐴 ≪ 훿 푀}. ii) 훿(푀) = ∩ {퐵: 퐵 ≤ 푀 푤푖푡ℎ 푀/퐵 푖푠 푠푖푛푔푢푙푎푟}. iii) 훿(푀) = ∩ {푘푒푟훷 ∶ 훷 ∈ 퐻표푚(푀, 푁) 푠푢푐ℎ 푡ℎ푎푡 푁 푖푠 푠푖푛푔푢푙푎푟 푠푖푚푝푙푒} 19 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.5, No.7, 2015 iv) 훿(푀) = ∩ {푘푒푟훷 ∶ 훷 ∈ 퐻표푚( 푀, 푁) 푠푢푐ℎ 푡ℎ푎푡 푁 푖푠 푠푖푛푔푢푙푎푟 푠푒푚푖 푠푖푛푔푢푙푎푟} Proposition:1.2. If f: M→N is an epimorphism with 푘푒푟푓 ≤ 훿(푀), 푡ℎ푒푛 훿(푁) = 푓(훿(푀)). Proof: [4]. Lemma:1.5. Let P be a small projective module, then 훿(푀) ≪훿 P. Proof: Let P be a small-projective module and 푃 = 훿(푃) + 푌, where P/Y is singular, by hypothesis P= A ⊕ B such that P M M f 0 A A ≤ Y and B∩Y ≤ δ(P), Then Y = A ⊕ (B∩Y) and so P = δ(P) ⊕A. since A is summand of P, there exists a sub module X ≤ δ(P) such that P = = X ⊕ A. Since 훿(푋) = 푋 ∩ 훿(푃) = 푋, X is semi simple projective and P/Y is epimorphic image of 푃/퐴 ≅ 푋 => 푃/푌 is projective and singular, we have P = Y. Hence δ(P) ≪δ P. Proposition:1.3. Let M and N be any R-modules. Then Following are equivalent: i) 푁 ≪훿 푀. ii) If X + N =M, then M = X⊕Y for projective semi simple sub module Y, with Y ≤ N. Proof: [4] Proposition:1.4. For Hollow module N the following conditions are equivalent: i) N is δ-small M-pseudo projective module. ii) N is M-pseudo projective module. Proof: (i) => (ii) is obvious. (ii)=> (i) Let R-module N be a M- pseudo projective module. A ≤ M, any small epimorphism M M f : N and natural epimorphism : M . For a sub module A of M, 퐴 ≪ 푀, then A A A 훿 is direct summand of M, there exists a decomposition M A B such that A N F and B N F ≪훿 푀, there exists a homomorphism h: N M such that the diagram N h f M M 0 A i.e. g.h f . Hence N is M-pseudo projective module.// Proposition:1.5. Let M be a δ-small M-Pseudo projective module, then following conditions are equivalent: i) M is S.F. If M/A is isomorphic to direct summand of M, A ≤ M.
Load more

Recommended publications
  • Topics in Module Theory
    Chapter 7 Topics in Module Theory This chapter will be concerned with collecting a number of results and construc- tions concerning modules over (primarily) noncommutative rings that will be needed to study group representation theory in Chapter 8. 7.1 Simple and Semisimple Rings and Modules In this section we investigate the question of decomposing modules into \simpler" modules. (1.1) De¯nition. If R is a ring (not necessarily commutative) and M 6= h0i is a nonzero R-module, then we say that M is a simple or irreducible R- module if h0i and M are the only submodules of M. (1.2) Proposition. If an R-module M is simple, then it is cyclic. Proof. Let x be a nonzero element of M and let N = hxi be the cyclic submodule generated by x. Since M is simple and N 6= h0i, it follows that M = N. ut (1.3) Proposition. If R is a ring, then a cyclic R-module M = hmi is simple if and only if Ann(m) is a maximal left ideal. Proof. By Proposition 3.2.15, M =» R= Ann(m), so the correspondence the- orem (Theorem 3.2.7) shows that M has no submodules other than M and h0i if and only if R has no submodules (i.e., left ideals) containing Ann(m) other than R and Ann(m). But this is precisely the condition for Ann(m) to be a maximal left ideal. ut (1.4) Examples. (1) An abelian group A is a simple Z-module if and only if A is a cyclic group of prime order.
    [Show full text]
  • Modular Representation Theory of Finite Groups Jun.-Prof. Dr
    Modular Representation Theory of Finite Groups Jun.-Prof. Dr. Caroline Lassueur TU Kaiserslautern Skript zur Vorlesung, WS 2020/21 (Vorlesung: 4SWS // Übungen: 2SWS) Version: 23. August 2021 Contents Foreword iii Conventions iv Chapter 1. Foundations of Representation Theory6 1 (Ir)Reducibility and (in)decomposability.............................6 2 Schur’s Lemma...........................................7 3 Composition series and the Jordan-Hölder Theorem......................8 4 The Jacobson radical and Nakayama’s Lemma......................... 10 Chapter5 2.Indecomposability The Structure of and Semisimple the Krull-Schmidt Algebras Theorem ...................... 1115 6 Semisimplicity of rings and modules............................... 15 7 The Artin-Wedderburn structure theorem............................ 18 Chapter8 3.Semisimple Representation algebras Theory and their of Finite simple Groups modules ........................ 2226 9 Linear representations of finite groups............................. 26 10 The group algebra and its modules............................... 29 11 Semisimplicity and Maschke’s Theorem............................. 33 Chapter12 4.Simple Operations modules on over Groups splitting and fields Modules............................... 3436 13 Tensors, Hom’s and duality.................................... 36 14 Fixed and cofixed points...................................... 39 Chapter15 5.Inflation, The Mackey restriction Formula and induction and Clifford................................ Theory 3945 16 Double cosets...........................................
    [Show full text]
  • Economic Indicators and Social Networks: New Approaches to Measuring Poverty, Prices, and Impacts of Technology
    Economic Indicators and Social Networks: New approaches to measuring poverty, prices, and impacts of technology by Niall Carrigan Keleher A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Information Management and Systems in the Graduate Division of the University of California, Berkeley Committee in charge: Dr. Joshua Evan Blumenstock, Chair Dr. John Chuang Dr. Jeremy Magruder Fall 2019 1 Abstract Economic Indicators and Social Networks: New approaches to measuring poverty, prices, and impacts of technology by Niall Carrigan Keleher Doctor of Philosophy in Information Management and Systems University of California, Berkeley Dr. Joshua Evan Blumenstock, Chair Collecting data to inform policy decisions is an ongoing global challenge. While some data collection has become routine, certain populations remain dicult to reach. From targeting social protection programs in densely-populated urban areas to reaching the “last mile” of infrastructure coverage, data collection and service delivery go hand-in-hand. Understanding the populations that live in urban communities as well as remote villages can help to tailor the design, targeting, and implementation of development programs. New sources of information have the potential to improve awareness of the needs and preferences of individuals, households, and communities. The goal of this dissertation is to provide multiple vantage points on the role that data, commu- nity input, and individual preferences can play in informing development policy. The empirical investigation presented in this dissertation covers two studies in Liberia and one in the Philippines. The unifying theme of the three chapters is the exploration of new sources of information about hard-to-reach populations.
    [Show full text]
  • Algebraic Modules for Finite Groups
    Algebraic Modules for Finite Groups David A. Craven January 2007 For the purposes of this talk, G is a finite group, K is an algebraically closed field of characteristic p, where p | |G|, and all modules are finite-dimensional. 1 Algebraic Modules The Green ring of KG-modules is defined to be the free abelian group on the basis set of all indecomposable KG-modules, with M + N defined to be equal to M ⊕ N, and the product of two modules defined as M ⊗ N. Notice that not all elements of the Green ring can be thought of as modules, since they could have negative multiplicities attached; they are virtual modules. The structure of the Green ring, while a commutative ring with a 1, is far from that of traditional commutative rings. For example, it is not an integral domain: in general, it has nilpotent elements. It is also in general infinite-dimensional. We can still, however, carry over some notions from algebraic number theory. One of those is algebraic modules. A module is said to be algebraic if it satisfies some polynomial equation in the Green ring, with co-efficients in Z. Proposition 1.1 Let M be a KG-module. Then the following are equivalent: (i) M is algebraic; (ii) M satisfies a monic polynomial equation in the Green ring with co-efficients in Z; and (iii) there are only finitely many different indecomposable summands of the (infinite-dimensional) module T (M) = M ⊕ M ⊗2 ⊕ M ⊗3 ⊕ · · · . The third equivalent condition is often the easiest to use in actually deciding if modules are algebraic or not.
    [Show full text]
  • Regular and Semisimple Modules
    Pacific Journal of Mathematics REGULAR AND SEMISIMPLE MODULES T. J. CHEATHAM AND J. R. SMITH Vol. 65, No. 2 October 1976 PACIFIC JOURNAL OF MATHEMATICS Vol. 65, No. 2, 1976 REGULAR AND SEMISIMPLE MODULES T. J. GHEATHAM AND J. R. SMITH A module is regular if all its submodules are (Cohn) pure. The family of all regular modules is closed under products if and only if RjJ{R) is a von Neumann regular ring. If each regular ϋJ-module is semisimple then R is a Γ-ring. An extra condition is needed for the converse* Character modules and extensions of regular and semisimple modules are investigated. 1* Introduction* Rings will be associative with identity and modules will be (left) unitary. R will denote a ring which is not assumed commutative unless specifically stated and J{R) will denote the Jacobson radical of R. Fieldhouse [5] calls a module B regular if each submodule A of B is pure in B, i.e., the inclusion 0—• A—>B remains exact upon tensor ing by any (right) iϋ-module. Regular modules have been studied under different definitions by Ware [12], Zelmanowitz [14], and Ramamurthi and Rangaswamy [9]. A module is semisimple if it is a sum of simple modules. For a subset A of a module J5, (0: A) will denote the left ideal {r eR\rx — 0 for all x e A}. 2* Products. The class of all semisimple modules is closed under products if and only if R/J(R) is a semisimple (Artinian) ring. This follows from the canonical embedding R/J(R) <=-+ ΠR/M, where the product is taken over the set of maximal left ideals M of R.
    [Show full text]
  • Li, Yiyang the Principal Indecomposable Modules of Sl3(K). J
    Li, Yiyang The principal indecomposable modules of sl3(k). J. Algebra Number Theory Appl. 25, No. 1, 45-58 (2012). Mathematics Subject Classification 2010: *17B10, 17B45, 17B50, 17B56 Keywords: Lie algebra of a reductive algebraic group; simple module; p-character; reduced universal enveloping algebra; standard Levi form; subregular nilpotent; baby Verma module; principal indecomposable module; self-extension Reviewer: J¨orgFeldvoss (8086) Let g be a finite dimensional restricted Lie algebra over an algebraically closed field k of prime characteristic p. Then for every simple g-module S there exists a linear form χ on g, the so-called p-character of S, such that S is a simple module for a certain finite dimensional quotient Uχ(g) of the universal enveloping algebra U(g) of g. In case g is the Lie algebra of a reductive algebraic group G, the linear dual g∗ and g are G-isomorphic and so the Jordan decomposition of elements in g transfers to g∗. Hence χ can be decomposed into a sum of its semisimple and nilpotent parts. It is well known that it is enough to consider only simple Uχ(g)-modules for nilpotent χ, but, in general, the structure of these simple modules is still unkown. If g = sln(k), then one can even assume that χ is in standard Levi form. In particular χ vanishes on the standard Borel subalgebra of g. The easiest case is when χ is regular, that is, when the dimension of the G- stabilizer of χ is equal to the rank of G. In this case every simple Uχ(g)-module is isomorphic to a baby Verma module, that is, a module induced from a one-dimensional (restricted) module for the standard Borel subalgebra of g (if χ is in standard Levi form).
    [Show full text]
  • Module (Mathematics) 1 Module (Mathematics)
    Module (mathematics) 1 Module (mathematics) In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring. Modules also generalize the notion of abelian groups, which are modules over the ring of integers. Thus, a module, like a vector space, is an additive abelian group; a product is defined between elements of the ring and elements of the module, and this multiplication is associative (when used with the multiplication in the ring) and distributive. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Motivation In a vector space, the set of scalars forms a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization. In commutative algebra, it is important that both ideals and quotient rings are modules, so that many arguments about ideals or quotient rings can be combined into a single argument about modules. In non-commutative algebra the distinction between left ideals, ideals, and modules becomes more pronounced, though some important ring theoretic conditions can be expressed either about left ideals or left modules. Much of the theory of modules consists of extending as many as possible of the desirable properties of vector spaces to the realm of modules over a "well-behaved" ring, such as a principal ideal domain.
    [Show full text]
  • Endomorphism Rings of Simple Modules Over Group Rings
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 4, April 1996 ENDOMORPHISM RINGS OF SIMPLE MODULES OVER GROUP RINGS ROBERT L. SNIDER (Communicated by Ken Goodearl) Abstract. If N is a finitely generated nilpotent group which is not abelian- by-finite, k afield,andDa finite dimensional separable division algebra over k, then there exists a simple module M for the group ring k[G] with endomor- phism ring D. An example is given to show that this cannot be extended to polycyclic groups. Let N be a finite nilpotent group and k a field. The Schur index of every irreducible representation of N is at most two. This means that for every irreducible module M of the group ring k[N], Endk[N](M) is either a field or the quaternion algebra over a finite field extension of k [6, p.564]. The purpose of this paper is to examine the situation when N is an infinite finitely generated nilpotent group. In this case, the endomorphism rings of simple modules are still finite dimensional [5, p.337]. However the situation is much different and in fact we prove Theorem 1. Let k be a field, N a finitely generated nilpotent group which is not abelian-by-finite, and D a separable division algebra finite dimensional over k, then there exists a simple k[N]-module M with Endk[N](M)=D. IdonotknowifDin Theorem 1 can be any inseparable division algebra. The calculations seem quite difficult for large inseparable division algebras although the calculations below would show that small inseparable algebras (where the center is a simple extension) do occur.
    [Show full text]
  • Review, Revise, and (Re)Release: Updating an Information Literacy Tutorial to Embed a Science Information Life Cycle
    Previous Contents Next Issues in Science and Technology Librarianship Fall 2013 DOI:10.5062/F4F769JQ Review, Revise, and (re)Release: Updating an Information Literacy Tutorial to Embed a Science Information Life Cycle Jeffra Diane Bussmann STEM/Web Librarian California State University, East Bay Hayward, California [email protected] Caitlin E. Plovnick Reference & Instruction Librarian University of California, Irvine Irvine, California [email protected] Copyright 2013, Jeffra Diane Bussmann and Caitlin E. Plovnick. Used with permission. Abstract In 2008, University of California, Irvine (UCI) Libraries launched their first Find Science Information online tutorial. It was an innovative web-based tool, containing not only informative content but also interactive activities, embedded hyperlinked resources, and reflective quizzes, all designed primarily to educate undergraduate science students on the basic components of finding science information. From its inception, the UCI Libraries intended to update the tutorial, ensuring its continued relevance and currency. This article covers the authors' process in evaluating and revising the tutorial, including usability testing and the changes made to the tutorial based on findings. One particular discovery was a need to contextualize the tutorial by more thoroughly addressing the scholarly communication system of the sciences, which led to its being restructured around a science information life cycle model. A literature review and comparison images from the original and revised tutorial are included. Introduction Convincing students of the importance of information literacy (IL) is a challenge that librarians usually address through classes, workshops, one-shot sessions, and online tutorials. We study theories and methods for activating this transformative knowledge in face-to-face and online environments.
    [Show full text]
  • Linear Representations of Finite Groups
    Linear Representations of Finite Groups Randall R. Holmes Auburn University 1 0 Introduction Let G be a finite group, let K be a field, and let V be a finite-dimensional vector space over K. Denote by GL(V ) the group of invertible linear transformations from V to itself. A group homomorphism ρ : G ! GL(V ) is called a linear K-representation of G in V (or just a representation of G for short). One gains information about the structure of G by studying the totality of representa- tions of G (i.e., various ρ, V , and K). Example. Suppose K = C. If every \irreducible" representation of G (that is, one admitting no proper \subrepresentation") is of the form ρ : G ! GL(V ) with dim V = 1, then G is abelian (and conversely). Here are some notable applications of representation theory: (1) (Burnside) If jGj = paqb (p, q prime), then G is solvable. (Proof given in Section 25.) (2) (Feit-Thompson) Every group of odd order is solvable. (3) Classification of Finite Simple Groups. (Proof uses both the \ordinary" theory (char K = 0) and the \modular" theory (char K = p, prime).) (4) Quantum mechanics. Let ρ : G ! GL(V ) be a representation. Define the associated character χ : G ! K by χ(a) = tr ρ(a). By passing from ρ to the associated character χ, one loses information in general, but enough information is retained to allow proofs of important results. For instance, the theorem of Burnside stated above uses only characters, not actual repre- sentations. Much of the power of character theory comes from its deep connections with number theory.
    [Show full text]
  • ENDOTRIVIAL MODULES for P-SOLVABLE GROUPS 1
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 363, Number 9, September 2011, Pages 4979–4996 S 0002-9947(2011)05307-9 Article electronically published on April 19, 2011 ENDOTRIVIAL MODULES FOR p-SOLVABLE GROUPS JON F. CARLSON, NADIA MAZZA, AND JACQUES THEVENAZ´ Abstract. We determine the torsion subgroup of the group of endotrivial modules for a finite solvable group in characteristic p. We also prove that our result would hold for p-solvable groups, provided a conjecture can be proved for the case of p-nilpotent groups. 1. Introduction In this paper, we analyse the group T (G) of endotrivial modules in characteris- tic p for a finite group G which is p-solvable. Our results are mainly concerned with the torsion subgroup TT(G)ofT (G). The description is not quite complete since it depends on a conjecture1 for the p-nilpotent case. However, we can prove the conjecture in the solvable case, hence we obtain final results for all solvable groups. Recall that endotrivial modules were introduced by Dade [9] and appear natu- rally in modular representation theory. Several contributions towards the general goal of classifying those modules have already been obtained. The most important result for our purposes is the classification of all endotrivial modules for p-groups ([6], [7], [8], [2]). Other results have been obtained in [4], [5], [3], [15], and [16]. We start with the analysis of T (G) for a finite p-nilpotent group G.LetP be a ∼ Sylow p-subgroup of G and N = Op (G)sothatG/N = P . We first show that ∼ T (G) = K(G) ⊕ T (P ) , where K(G) is the kernel of the restriction to P .
    [Show full text]
  • The Development of a SOA and Apis Based Chinese Cloud Cataloguing Platform
    大學圖書館 19 卷 1 期（民國 104 年 3 月） 頁 21-40 The Development of a SOA and APIs Based Chinese Cloud Cataloguing Platform 張迺貞 Naicheng Chang 大同大學通識教育中心副教授 Associate Professor, General Education Center, Tatung University E-mail：[email protected] 蔡育欽 Yuchin Tsai 國家實驗研究院國家網路與計算中心副研究員 Associate Researcher, National Center for High-performance Computing, NARL, NSC E-mail：[email protected] Alan Hopkinson Chairman of the Permanent UNIMARC Committee of IFLA E-mail：[email protected] 【Abstract】 The purpose of this project is to build the first Chinese cloud resources pooling platform with an on-demand self-service infrastructure for libraries and Library Management System (LMS) vendors by developing a Service-Oriented Architecture (SOA) and Application Programming Interfaces (APIs) procedure to access heterogeneous library bibliographic records. It aims to provide a cloud computing cataloguing service to libraries, academic and LMS vendors in Taiwan for future use. The system platform design was developed under two stages: 1) constructing the cloud computing infrastructure includes setting up hardware facilities, integrating Internet resources and using a set of widely accepted free or open source virtual technologies, 2) deploying the cataloguing technology includes the development of software, using developers and a team of library professionals to reduce the technology gap and to ensure that library needs are taken into account. The project demonstrates a potentially workable process in terms of availability and feasibility of a Chinese library cataloguing platform that is based on cloud computing 收稿日期：民國 103 年 10 月 8 日；接受日期：民國 103 年 12 月 1218 日 doi:10.6146/univj.19-1.02 張迺貞、蔡育欽、Alan Hopkinson infrastructure.
    [Show full text]