Groups with Identical Subgroup Lattices in All Powers
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ORTHOGONAL GROUP of CERTAIN INDEFINITE LATTICE Chang Heon Kim* 1. Introduction Given an Even Lattice M in a Real Quadratic Space
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 20, No. 1, March 2007 ORTHOGONAL GROUP OF CERTAIN INDEFINITE LATTICE Chang Heon Kim* Abstract. We compute the special orthogonal group of certain lattice of signature (2; 1). 1. Introduction Given an even lattice M in a real quadratic space of signature (2; n), Borcherds lifting [1] gives a multiplicative correspondence between vec- tor valued modular forms F of weight 1¡n=2 with values in C[M 0=M] (= the group ring of M 0=M) and meromorphic modular forms on complex 0 varieties (O(2) £ O(n))nO(2; n)=Aut(M; F ). Here NM denotes the dual lattice of M, O(2; n) is the orthogonal group of M R and Aut(M; F ) is the subgroup of Aut(M) leaving the form F stable under the natural action of Aut(M) on M 0=M. In particular, if the signature of M is (2; 1), then O(2; 1) ¼ H: O(2) £ O(1) and Borcherds' theory gives a lifting of vector valued modular form of weight 1=2 to usual one variable modular form on Aut(M; F ). In this sense in order to work out Borcherds lifting it is important to ¯nd appropriate lattice on which our wanted modular group acts. In this article we will show: Theorem 1.1. Let M be a 3-dimensional even lattice of all 2 £ 2 integral symmetric matrices, that is, ½µ ¶ ¾ AB M = j A; B; C 2 Z BC Received December 30, 2006. 2000 Mathematics Subject Classi¯cation: Primary 11F03, 11H56. -
ON the SHELLABILITY of the ORDER COMPLEX of the SUBGROUP LATTICE of a FINITE GROUP 1. Introduction We Will Show That the Order C
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 7, Pages 2689{2703 S 0002-9947(01)02730-1 Article electronically published on March 12, 2001 ON THE SHELLABILITY OF THE ORDER COMPLEX OF THE SUBGROUP LATTICE OF A FINITE GROUP JOHN SHARESHIAN Abstract. We show that the order complex of the subgroup lattice of a finite group G is nonpure shellable if and only if G is solvable. A by-product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups. 1. Introduction We will show that the order complex of the subgroup lattice of a finite group G is (nonpure) shellable if and only if G is solvable. The proof of nonshellability in the nonsolvable case involves the determination of the homotopy type of the order complexes of the subgroup lattices of many minimal simple groups. We begin with some history and basic definitions. It is assumed that the reader is familiar with some of the rudiments of algebraic topology and finite group theory. No distinction will be made between an abstract simplicial complex ∆ and an arbitrary geometric realization of ∆. Maximal faces of a simplicial complex ∆ will be called facets of ∆. Definition 1.1. A simplicial complex ∆ is shellable if the facets of ∆ can be ordered σ1;::: ,σn so that for all 1 ≤ i<k≤ n thereexistssome1≤ j<kand x 2 σk such that σi \ σk ⊆ σj \ σk = σk nfxg. The list σ1;::: ,σn is called a shelling of ∆. -
7 LATTICE POINTS and LATTICE POLYTOPES Alexander Barvinok
7 LATTICE POINTS AND LATTICE POLYTOPES Alexander Barvinok INTRODUCTION Lattice polytopes arise naturally in algebraic geometry, analysis, combinatorics, computer science, number theory, optimization, probability and representation the- ory. They possess a rich structure arising from the interaction of algebraic, convex, analytic, and combinatorial properties. In this chapter, we concentrate on the the- ory of lattice polytopes and only sketch their numerous applications. We briefly discuss their role in optimization and polyhedral combinatorics (Section 7.1). In Section 7.2 we discuss the decision problem, the problem of finding whether a given polytope contains a lattice point. In Section 7.3 we address the counting problem, the problem of counting all lattice points in a given polytope. The asymptotic problem (Section 7.4) explores the behavior of the number of lattice points in a varying polytope (for example, if a dilation is applied to the polytope). Finally, in Section 7.5 we discuss problems with quantifiers. These problems are natural generalizations of the decision and counting problems. Whenever appropriate we address algorithmic issues. For general references in the area of computational complexity/algorithms see [AB09]. We summarize the computational complexity status of our problems in Table 7.0.1. TABLE 7.0.1 Computational complexity of basic problems. PROBLEM NAME BOUNDED DIMENSION UNBOUNDED DIMENSION Decision problem polynomial NP-hard Counting problem polynomial #P-hard Asymptotic problem polynomial #P-hard∗ Problems with quantifiers unknown; polynomial for ∀∃ ∗∗ NP-hard ∗ in bounded codimension, reduces polynomially to volume computation ∗∗ with no quantifier alternation, polynomial time 7.1 INTEGRAL POLYTOPES IN POLYHEDRAL COMBINATORICS We describe some combinatorial and computational properties of integral polytopes. -
INTEGER POINTS and THEIR ORTHOGONAL LATTICES 2 to Remove the Congruence Condition
INTEGER POINTS ON SPHERES AND THEIR ORTHOGONAL LATTICES MENNY AKA, MANFRED EINSIEDLER, AND URI SHAPIRA (WITH AN APPENDIX BY RUIXIANG ZHANG) Abstract. Linnik proved in the late 1950’s the equidistribution of in- teger points on large spheres under a congruence condition. The congru- ence condition was lifted in 1988 by Duke (building on a break-through by Iwaniec) using completely different techniques. We conjecture that this equidistribution result also extends to the pairs consisting of a vector on the sphere and the shape of the lattice in its orthogonal complement. We use a joining result for higher rank diagonalizable actions to obtain this conjecture under an additional congruence condition. 1. Introduction A theorem of Legendre, whose complete proof was given by Gauss in [Gau86], asserts that an integer D can be written as a sum of three squares if and only if D is not of the form 4m(8k + 7) for some m, k N. Let D = D N : D 0, 4, 7 mod8 and Z3 be the set of primitive∈ vectors { ∈ 6≡ } prim in Z3. Legendre’s Theorem also implies that the set 2 def 3 2 S (D) = v Zprim : v 2 = D n ∈ k k o is non-empty if and only if D D. This important result has been refined in many ways. We are interested∈ in the refinement known as Linnik’s problem. Let S2 def= x R3 : x = 1 . For a subset S of rational odd primes we ∈ k k2 set 2 D(S)= D D : for all p S, D mod p F× . -
The M\" Obius Number of the Socle of Any Group
Contents 1 Definitions and Statement of Main Result 2 2 Proof of Main Result 3 2.1 Complements ........................... 3 2.2 DirectProducts.......................... 5 2.3 The Homomorphic Images of a Product of Simple Groups . 6 2.4 The M¨obius Number of a Direct Power of an Abelian Simple Group ............................... 7 2.5 The M¨obius Number of a Direct Power of a Nonabelian Simple Group ............................... 8 2.6 ProofofTheorem1........................ 9 arXiv:1002.3503v1 [math.GR] 18 Feb 2010 1 The M¨obius Number of the Socle of any Group Kenneth M Monks Colorado State University February 18, 2010 1 Definitions and Statement of Main Result The incidence algebra of a poset P, written I (P ) , is the set of all real- valued functions on P × P that vanish for ordered pairs (x, y) with x 6≤ y. If P is finite, by appropriately labeling the rows and columns of a matrix with the elements of P , we can see the elements of I (P ) as upper-triangular matrices with zeroes in certain locations. One can prove I (P ) is a subalgebra of the matrix algebra (see for example [6]). Notice a function f ∈ I(P ) is invertible if and only if f (x, x) is nonzero for all x ∈ P , since then we have a corresponding matrix of full rank. A natural function to consider that satisfies this property is the incidence function ζP , the characteristic function of the relation ≤P . Clearly ζP is invertible by the above criterion, since x ≤ x for all x ∈ P . We define the M¨obius function µP to be the multiplicative inverse of ζP in I (P ) . -
(Ipad) Subgroups Generated by a Subset of G Cyclic Groups Suppo
Read: Dummit and Foote Chapter 2 Lattice of subgroups of a group (iPad) Subgroups generated by a subset of G Cyclic groups Suppose is a subgroup of G. We will say that is the smallest subgroup of G containing S if S and if H is a subgroup of G containing S then S. Lemma 1. If and 0 both satisfy this denition then = 0. (Proof below that S = S .) h i hh ii Theorem 2. Let G be a group and S a subset. Then there is a unique smallest subgroup of G containing S. First proof (not constructive). Let S = H: h i H a subgroup of G S\H This is the unique smallest subgroup of G containing S. Indeed, these are all subgroups so their intersection is a subgroup: if x; y S then x; y H for all H in the intersection, so 1 2 h i 2 1 xy H and x¡ H and 1 H for all these subgroups, so xy; x¡ ; 1 S proving S is a gro2up. 2 2 2 h i h i Moreover, S is the smallest subgroup of G containing S because if H is any subgroup of G containinhg Si then H is one of the groups in the intersection and so S H. h i Second proof: Let S be dened as the set of all products (words) hh ii a a a 1 2 N 1 where either ai S or ai¡ S and if N = 0 the empty product is interpreted as 1G. I claim this is the smal2lest subgro2up of G containing S. -
On the Structure of Overgroup Lattices in Exceptional Groups of Type G2(Q)
On the Structure of Overgroup Lattices in Exceptional Groups of Type G2(q) Jake Wellens Mentor: Michael Aschbacher December 4, 2017 Abstract: We study overgroup lattices in groups of type G2 over finite fields and reduce Shareshi- ans conjecture on D∆-lattices in the G2 case to a smaller question about the number of maximal elements that can be conjugate in such a lattice. Here we establish that, modulo this smaller ques- tion, no sufficiently large D∆-lattice is isomorphic to the lattice of overgroups OG(H) in G = G2(q) for any prime power q and any H ≤ G. This has application to the question of Palfy and Pud- lak as to whether each finite lattice is isomorphic to some overgroup lattice inside some finite group. 1 Introduction Recall that a lattice is a partially ordered set in which each element has a greatest lower bound and a least upper bound. In 1980, Palfy and Pudlak [PP] proved that the following two statements are equivalent: (1) Every finite lattice is isomorphic to an interval in the lattice of subgroups of some finite group. (2) Every finite lattice is isomorphic to the lattice of congruences of some finite algebra. This leads one to ask the following question: The Palfy-Pudlak Question: Is each finite lattice isomorphic to a lattice OG(H) of overgroups of some subgroup H in a finite group G? There is strong evidence to believe that the answer to Palfy-Pudlak Question is no. While it would suffice to exhibit a single counterexample, John Shareshian has conjectured that an entire class of lattices - which he calls D∆-lattices - are not isomorphic to overgroup lattices in any finite groups. -
Groups with Almost Modular Subgroup Lattice Provided by Elsevier - Publisher Connector
Journal of Algebra 243, 738᎐764Ž. 2001 doi:10.1006rjabr.2001.8886, available online at http:rrwww.idealibrary.com on View metadata, citation and similar papers at core.ac.uk brought to you by CORE Groups with Almost Modular Subgroup Lattice provided by Elsevier - Publisher Connector Francesco de Giovanni and Carmela Musella Dipartimento di Matematica e Applicazioni, Uni¨ersita` di Napoli ‘‘Federico II’’, Complesso Uni¨ersitario Monte S. Angelo, Via Cintia, I 80126, Naples, Italy and Yaroslav P. Sysak1 Institute of Mathematics, Ukrainian National Academy of Sciences, ¨ul. Tereshchenki¨ska 3, 01601 Kie¨, Ukraine Communicated by Gernot Stroth Received November 14, 2000 DEDICATED TO BERNHARD AMBERG ON THE OCCASION OF HIS 60TH BIRTHDAY 1. INTRODUCTION A subgroup of a group G is called modular if it is a modular element of the lattice ᑦŽ.G of all subgroups of G. It is clear that everynormal subgroup of a group is modular, but arbitrarymodular subgroups need not be normal; thus modularitymaybe considered as a lattice generalization of normality. Lattices with modular elements are also called modular. Abelian groups and the so-called Tarski groupsŽ i.e., infinite groups all of whose proper nontrivial subgroups have prime order. are obvious examples of groups whose subgroup lattices are modular. The structure of groups with modular subgroup lattice has been described completelybyIwasawa wx4, 5 and Schmidt wx 13 . For a detailed account of results concerning modular subgroups of groups, we refer the reader towx 14 . 1 This work was done while the third author was visiting the Department of Mathematics of the Universityof Napoli ‘‘Federico II.’’ He thanks the G.N.S.A.G.A. -
On the Lattice of Subgroups of a Free Group: Complements and Rank
journal of Groups, Complexity, Cryptology Volume 12, Issue 1, 2020, pp. 1:1–1:24 Submitted Sept. 11, 2019 https://gcc.episciences.org/ Published Feb. 29, 2020 ON THE LATTICE OF SUBGROUPS OF A FREE GROUP: COMPLEMENTS AND RANK JORDI DELGADO AND PEDRO V. SILVA Centro de Matemática, Universidade do Porto, Portugal e-mail address: [email protected] Centro de Matemática, Universidade do Porto, Portugal e-mail address: [email protected] Abstract. A ∨-complement of a subgroup H 6 Fn is a subgroup K 6 Fn such that H ∨ K = Fn. If we also ask K to have trivial intersection with H, then we say that K is a ⊕-complement of H. The minimum possible rank of a ∨-complement (resp., ⊕-complement) of H is called the ∨-corank (resp., ⊕-corank) of H. We use Stallings automata to study these notions and the relations between them. In particular, we characterize when complements exist, compute the ∨-corank, and provide language-theoretical descriptions of the sets of cyclic complements. Finally, we prove that the two notions of corank coincide on subgroups that admit cyclic complements of both kinds. 1. Introduction Subgroups of free groups are complicated. Of course not the structure of the subgroups themselves (which are always free, a classic result by Nielsen and Schreier) but the relations between them, or more precisely, the lattice they constitute. A first hint in this direction is the fact that (free) subgroups of any countable rank appear as subgroups of the free group of rank 2 (and hence of any of its noncyclic subgroups) giving rise to a self-similar structure. -
The Theory of Lattice-Ordered Groups
The Theory ofLattice-Ordered Groups Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 307 The Theory of Lattice-Ordered Groups by V. M. Kopytov Institute ofMathematics, RussianAcademyof Sciences, Siberian Branch, Novosibirsk, Russia and N. Ya. Medvedev Altai State University, Bamaul, Russia Springer-Science+Business Media, B.Y A C.I.P. Catalogue record for this book is available from the Library ofCongress. ISBN 978-90-481-4474-7 ISBN 978-94-015-8304-6 (eBook) DOI 10.1007/978-94-015-8304-6 Printed on acid-free paper All Rights Reserved © 1994 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1994. Softcover reprint ofthe hardcover Ist edition 1994 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrie val system, without written permission from the copyright owner. Contents Preface IX Symbol Index Xlll 1 Lattices 1 1.1 Partially ordered sets 1 1.2 Lattices .. ..... 3 1.3 Properties of lattices 5 1.4 Distributive and modular lattices. Boolean algebras 6 2 Lattice-ordered groups 11 2.1 Definition of the l-group 11 2.2 Calculations in I-groups 15 2.3 Basic facts . 22 3 Convex I-subgroups 31 3.1 The lattice of convex l-subgroups .......... .. 31 3.2 Archimedean o-groups. Convex subgroups in o-groups. 34 3.3 Prime subgroups 39 3.4 Polars ... ..................... 43 3.5 Lattice-ordered groups with finite Boolean algebra of polars ...................... -
Finite Coxeter Groups and Their Subgroup Lattices
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 101, 82-94 (1986) Finite Coxeter Groups and Their Subgroup Lattices TOHRUUZAWA * Department of Mathematics, University of Tokyo, Tokyo, 113, Japan Communicated by Walter Feit Received May 4, 1984 INTRODUCTION Galois theory states that if we draw the Hasse diagram of the lattice of intermediary fields between a field and one of its of Galois extensions, and look at it upside down, then we have the subgroup lattice of the Galois group of the extension. Once we have this interpretation of Galois theory in mind, it is quite hard to resist the temptation to attach a group to a non-Galois extension. For example, one might attach the cyclic group of order 4 to the extension Q(2”4)/Q. So if we try to consider the possibility of a more general “Galois theory” or the possibility of a covariant Galois theory (which is in general not possible), we are inevitably led to the investigation of the relation of the structure of a group and its subgroup lattice. Investigation on (roughly) this line commenced with A. Rottlaender, a student of I. Schur, in 1928. We shall show in this note that a nice family of groups is determined by its subgroup lattices. Before explaining our main result, let us fix some notations. Let G be a group. Then by L(G) we denote the set of subgroups of G. This becomes a lattice with the usual set theoretic inclusion relation. -
On the Lattice Structure of Cyclic Groups of Order the Product of Distinct Primes
American Journal of Mathematics and Statistics 2018, 8(4): 96-98 DOI: 10.5923/j.ajms.20180804.03 On the Lattice Structure of Cyclic Groups of Order the Product of Distinct Primes Rosemary Jasson Nzobo1,*, Benard Kivunge2, Waweru Kamaku3 1Pan African University Institute for Basic Sciences, Technology and Innovation, Nairobi, Kenya 2Department of Mathematics, Kenyatta University, Nairobi, Kenya 3Pure and Applied Mathematics Department, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya Abstract In this paper, we give general formulas for counting the number of levels, subgroups at each level and number of ascending chains of subgroup lattice of cyclic groups of order the product of distinct primes. We also give an example to illustrate the concepts introduced in this work. Keywords Lattice, Cyclic Group, Subgroups, Chains it is isomorphic to for primes . In this paper, we present general formulas for finding the 1. Introduction 푛푚 number of levels, subgroupsℤ at each푛 ≠level푚 and number of A subgroup lattice is a diagram that includes all the ascending chains of cyclic groups of order … subgroups of the group and then connects a subgroup H at where are distinct primes. one level to a subgroup K at a higher level with a sequence 푝1푝2푝3 푝푚 of line segments if and only if H is a proper subgroup of K 푝푖 [1]. The study of subgroup lattice structures is traced back 2. Preliminaries from the first half of 20th century. For instance in 1953, Suzuki presented the extent to which a group is determined Definition 2.1 ([6]) Let L be a non empty set and < be a by its subgroup lattice in [2].