DOCSLIB.ORG

GENERAL THEOREMS APPLYING to ALL the GROUPS of ORDER 32 by G

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
GENERAL THEOREMS APPLYING to ALL the GROUPS of ORDER 32 by G 112 MA THEMA TICS: G. A. MILLER PROC. N. A. S. GENERAL THEOREMS APPLYING TO ALL THE GROUPS OF ORDER 32 By G. A. MILLER DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS Communicated December 26, 1935 The groups of low orders were at first usually determined by tentative methods but as group theory has been more generally developed more and more of these groups could also be derived as special cases of general the- orems. It was noted recently that the fourteen possible groups of order 16 could be thus determined and in the present article we aim to direct attention to general theorems by means of which it is possible to determine all the possible groups of order 32. We shall thus be able not only to throw new light on this somewhat complex system of groups but also pre- pare the way to determine all the groups of certain higher orders by similar methods. The methods which are emphasized here are based on the de- termination of groups by means of the subgroups contained therein which are assumed to be generated in each case by the squares of the operators contained in the group. If the squares of the operators of a group of order 2m generate a sub- group of index 2 the group is known to be cyclic since the squares of the operators of such a group generate its +-subgroup but when these squares generate a subgroup of index 4 comparatively little is known in regard to these groups except that each of them contains three and only three sub- groups of index 2. When m = 5, this subgroup of index 4 is abelian in view of the theorem that if a group involves less than nine operators which are generated by the squares of its operators then these squares are rela- tively commutative. The groups generated by the squares of the opera- tors of a non-cyclic group of order 32 are therefore, besides the identity and the group of order 2, the two groups of order 4 and the three abelian groups of order 8. The non-cyclic groups of order 32 may therefore be divided into seven categories according to the subgroups generated by the squares of their operators. It has been known for a long time that there is one and only one group of order 2m for every positive integral value of m, which satisfies the condition that the squares of all its operators are equal to the identity, and it has re- cently been established that the number of the groups which satisfy the con- dition, that the squares of their operators are the group of order 2 is 3(n -1) /2 when m is odd and (3m -4)/2 when m is even.1 It has been known for a long time that there are five and only five groups of order 2m, m > 3, which separately satisfy the condition that the squares of their operators generate a cyclic subgroup of index 4 since such a group is non-cyclic and Downloaded by guest on October 1, 2021 VOL. 22, 1936 MA THEMA TICS: G. A. MILLER 113 involves a cyclic subgroup of index 2. The total number of the groups of order 32 which result as special cases of the noted general theorems is 13. Hence it remains to determine general theorems relating to the groups of order 32 whose squared operators generate one of the following four groups: The four groups, the cyclic group of order 4, the abelian group of order 8 and of type 2,1 and the abelian group of this order and of type2 13. Excluding direct products in which one of the factors is the abelian group of type 1" there is one and only one abelian group which is determined by a given subgroup generated by the squares of its operators. Hence there are three abelian groups of order 32 which have one of the given four groups as the subgroup generated by its squares and we shall confine our attention in what follows to the non-abelian groups of order 32 which have one of the given four groups as the group generated by the squares of their operators. We begin with the case when these squares generate the four group and we shall at first confine our attention to the special case when the commu- tator subgroup of such a group is of order 2. Each of these groups con- tains a characteristic subgroup of index 2 generated by its operators of order 2 and those of its operators of order 4 (if any) whose squares are the commutator of order 2. This characteristic subgroup therefore belongs to one of the systems of groups which are determined by the fact that the squares of the operators of each group either constitute the identity or the group of order 2. Since the four group is in the central of every group which has the prop- erty that the squares of the operators of the group are the four group it re- sults that the given subgroup of index 2 contains at least three invariant operators of order 2. When this subgroup is abelian and of type 1' there is one and only one such group of order 2", m > 3. If it is of type 2, 1" there is one group of order 16, and there are two groups of order 2m, m > 4. In general, if it belongs to any one of the three well-known infinite systems of groups whose operators have two and only two distinct squares, there is one group whenever this subgroup is the direct product of such a group hav- ing a central of order 2 and the group of order 2 and there are two such groups whenever the second of the factor groups is the four group. Hence there is one such group of order 32 for each subgroup in the two of the given infinite systems characterized by the octic group and.the quaternion group, respectively, and there are two such groups in each of these systems when- ever m = 6. Therefore, the total number of these groups of order 2m in which the commutator subgroup is of order 2 is 3m -10, and there are 5 such groups of order 32. When both the subgroup generated by the squares of the operators of a group G and its commutator subgroup is the four group then G contains an abelian subgroup H of index 2. As the order of this subgroup is at least 16 the order of G is at least 32 and there is one and only one such G Downloaded by guest on October 1, 2021 114 1MA THEMA TICS: G. A. MILLER PRtOC. N. A. S. when His of type 14. When His of type 2, 12 there are three such groups of order 32. Finally, when H is of type 22 there are two groups in which every operator of H is transformed into its inverse. There are also two groups in which some but not all of these operators are transformed into their inverses and there is one group in which none of these operators is transformed into its inverse. There are therefore nine groups of order 32 which have the four group both for their commutator subgroup and also for the group generated by the squares of their operators. Since there are five other non-abelian groups of order 32 which have the four group as the group generated by the squares of their operators the total number of these non-abelian groups is fourteen. To determine the non-abelian groups of order 32 which have the cyclic group of order 4, as the subgroup generated by the squares of their opera- tors we may first consider those under which the operators of this cyclic group are invariant. There are two such groups in view of the general theorem that there are m-a groups of order 2m which have the property that the squares of their operators generate the cyclic group of order 2a and that the operators of this cyclic group are invariant under the group. One and only one of these m-a groups is obviously abelian. When the operators of order 4 in this cyclic subgroup are transformed into their in- verses under the group there is a subgroup of index 2 under which they are invariant. When this subgroup is abelian there are two groups in which each of its operators is transformed into its inverse and there is one group in which each of these operators is transformed into its third power. When the central is cyclic there is one more group so that there are four groups in which the given characteristic subgroup of index 2 is abelian. When this subgroup is non-abelian there are two such groups so that there are nine groups of order 32 which have the cyclic group of order 4 as the group gen- erated by the squares of their operators. If a group of order 2' has the abelian group of type 2,1 for the group of its squares it involves either the abelian group of type 3,1 or the abelian group of type 22 as an invariant subgroup. If such a G is of order 32 and involves the former of these abelian subgroups there is one G in which each operator of this subgroup is transformed into its inverse and there is also one group in which each of these operators is transformed into its third power, while there are two additional groups in which the cyclic subgroups of order 8 in H are interchanged.
Load more

Recommended publications
  • Categories, Functors, and Natural Transformations I∗
    Lecture 2: Categories, functors, and natural transformations I∗ Nilay Kumar June 4, 2014 (Meta)categories We begin, for the moment, with rather loose definitions, free from the technicalities of set theory. Definition 1. A metagraph consists of objects a; b; c; : : :, arrows f; g; h; : : :, and two operations, as follows. The first is the domain, which assigns to each arrow f an object a = dom f, and the second is the codomain, which assigns to each arrow f an object b = cod f. This is visually indicated by f : a ! b. Definition 2. A metacategory is a metagraph with two additional operations. The first is the identity, which assigns to each object a an arrow Ida = 1a : a ! a. The second is the composition, which assigns to each pair g; f of arrows with dom g = cod f an arrow g ◦ f called their composition, with g ◦ f : dom f ! cod g. This operation may be pictured as b f g a c g◦f We require further that: composition is associative, k ◦ (g ◦ f) = (k ◦ g) ◦ f; (whenever this composition makese sense) or diagrammatically that the diagram k◦(g◦f)=(k◦g)◦f a d k◦g f k g◦f b g c commutes, and that for all arrows f : a ! b and g : b ! c, we have 1b ◦ f = f and g ◦ 1b = g; or diagrammatically that the diagram f a b f g 1b g b c commutes. ∗This talk follows [1] I.1-4 very closely. 1 Recall that a diagram is commutative when, for each pair of vertices c and c0, any two paths formed from direct edges leading from c to c0 yield, by composition of labels, equal arrows from c to c0.
    [Show full text]
  • GROUPS with MINIMAX COMMUTATOR SUBGROUP Communicated by Gustavo A. Fernández-Alcober 1. Introduction a Group G Is Said to Have
    International Journal of Group Theory ISSN (print): 2251-7650, ISSN (on-line): 2251-7669 Vol. 3 No. 1 (2014), pp. 9-16. c 2014 University of Isfahan www.theoryofgroups.ir www.ui.ac.ir GROUPS WITH MINIMAX COMMUTATOR SUBGROUP F. DE GIOVANNI∗ AND M. TROMBETTI Communicated by Gustavo A. Fern´andez-Alcober Abstract. A result of Dixon, Evans and Smith shows that if G is a locally (soluble-by-finite) group whose proper subgroups are (finite rank)-by-abelian, then G itself has this property, i.e. the commutator subgroup of G has finite rank. It is proved here that if G is a locally (soluble-by-finite) group whose proper subgroups have minimax commutator subgroup, then also the commutator subgroup G0 of G is minimax. A corresponding result is proved for groups in which the commutator subgroup of every proper subgroup has finite torsion-free rank. 1. Introduction A group G is said to have finite rank r if every finitely generated subgroup of G can be generated by at most r elements, and r is the least positive integer with such property. It follows from results of V.V. Belyaev and N.F. Sesekin [1] that if G is any (generalized) soluble group of infinite rank whose proper subgroups are finite-by-abelian, then also the commutator subgroup G0 of G is finite. More recently, M.R. Dixon, M.J. Evans and H. Smith [4] proved that if G is a locally (soluble-by-finite) group in which the commutator subgroup of every proper subgroup has finite rank, then G0 has likewise finite rank.
    [Show full text]
  • Math 594: Homework 4
    Math 594: Homework 4 Due February 11, 2015 First Exam Feb 20 in class. 1. The Commutator Subgroup. Let G be a group. The commutator subgroup C (also denoted [G; G]) is the subgroup generated by all elements of the form g−1h−1gh. a). Prove that C is a normal subgroup of G, and that G=C is abelian (called the abelization of G). b). Show that [G; G] is contained in any normal subgroup N whose quotient G=N is abelian. State and prove a universal property for the abelianization map G ! G=[G; G]. c). Prove that if G ! H is a group homomorphism, then the commutator subgroup of G is taken to the commutator subgroup of H. Show that \abelianization" is a functor from groups to abelian groups. 2. Solvability. Define a (not-necessarily finite) group G to be solvable if G has a normal series feg ⊂ G1 ⊂ G2 · · · ⊂ Gt−1 ⊂ Gt = G in which all the quotients are abelian. [Recall that a normal series means that each Gi is normal in Gi+1.] a). For a finite group G, show that this definition is equivalent to the definition on problem set 3. b*). Show that if 1 ! A ! B ! C ! 1 is a short exact sequence of groups, then B if solvable if and only if both A and C are solvable. c). Show that G is solvable if and only if eventually, the sequence G ⊃ [G; G] ⊃ [[G; G]; [G; G]] ⊃ ::: terminates in the trivial group. This is called the derived series of G.
    [Show full text]
  • The Center and the Commutator Subgroup in Hopfian Groups
    The center and the commutator subgroup in hopfian groups 1~. I-hRSHON Polytechnic Institute of Brooklyn, Brooklyn, :N.Y., U.S.A. 1. Abstract We continue our investigation of the direct product of hopfian groups. Through- out this paper A will designate a hopfian group and B will designate (unless we specify otherwise) a group with finitely many normal subgroups. For the most part we will investigate the role of Z(A), the center of A (and to a lesser degree also the role of the commutator subgroup of A) in relation to the hopficity of A • B. Sections 2.1 and 2.2 contain some general results independent of any restric- tions on A. We show here (a) If A X B is not hopfian for some B, there exists a finite abelian group iv such that if k is any positive integer a homomorphism 0k of A • onto A can be found such that Ok has more than k elements in its kernel. (b) If A is fixed, a necessary and sufficient condition that A • be hopfian for all B is that if 0 is a surjective endomorphism of A • B then there exists a subgroup B. of B such that AOB-=AOxB.O. In Section 3.1 we use (a) to establish our main result which is (e) If all of the primary components of the torsion subgroup of Z(A) obey the minimal condition for subgroups, then A • is hopfian, In Section 3.3 we obtain some results for some finite groups B. For example we show here (d) If IB[ -~ p'q~.., q? where p, ql q, are the distinct prime divisors of ]B[ and ff 0 <e<3, 0~e~2 and Z(A) has finitely many elements of order p~ then A • is hopfian.
    [Show full text]
  • Automorphism Group of the Commutator Subgroup of the Braid Group
    ANNALES DE LA FACULTÉ DES SCIENCES Mathématiques STEPAN YU.OREVKOV Automorphism group of the commutator subgroup of the braid group Tome XXVI, no 5 (2017), p. 1137-1161. <http://afst.cedram.org/item?id=AFST_2017_6_26_5_1137_0> © Université Paul Sabatier, Toulouse, 2017, tous droits réservés. L’accès aux articles de la revue « Annales de la faculté des sci- ences de Toulouse Mathématiques » (http://afst.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://afst.cedram. org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Annales de la faculté des sciences de Toulouse Volume XXVI, no 5, 2017 pp. 1137-1161 Automorphism group of the commutator subgroup of the braid group (∗) Stepan Yu. Orevkov (1) 0 ABSTRACT.— Let Bn be the commutator subgroup of the braid 0 group Bn. We prove that Aut(Bn) = Aut(Bn) for n > 4. This answers a question asked by Vladimir Lin. 0 RÉSUMÉ.— Soit Bn le groupe dérivé du groupe de tresses Bn. On 0 montre que Aut(Bn) = Aut(Bn) pour n > 4, ce qui répond à une question posée par Vladimir Lin. 1. Introduction Let Bn be the braid group with n strings. It is generated by σ1, . , σn−1 (called standard or Artin generators) subject to the relations σiσj = σjσi for |i − j| > 1 ; σiσjσi = σjσiσj for |i − j| = 1.
    [Show full text]
  • The Structure of the Abelian Groups F,/(F,, F’)
    JOURNAL OF ALGEBRA 40, 301-308 (1976) The Structure of the Abelian Groups F,/(F,, f’) DENNIS SPELLMAN Fact&ad de C&&as, Departamento de Matemdticas, Universidad de Los Andes, M&da-Ed0 M&da, Venezuela Communicated by Marshall Hall, Jr. Received November 14, 1974 Let F be a free group of rank k with 2 < k < tu. Let n > 2 be a positive integer also. Let F, be the nth term in the lower central series of F and F’ the commutator subgroup of F. Then (F,, , F’) Q F, and the quotient group F,,/(F,, , F’) is abelian. The two main results are: (1) Fn/(Fn ,F’) zz T x A, where T is a finite abelian group and A is free abelian of countably infinite rank. (2) If n = 2, 3, or 4 and 2 < k < cc is arbitrary, or if n = 5 and k = 2, then F,/(F, , F’) is free abelian of countably infinite rank. I do not know whether F,/(F, , F’) is torsion free for arbitrary values of n. (I) DEFINITIONS AND NOTATION Let G be a group and a, b E G. Then (a, b) = a-W%& If S Z G, then gp(S) is the subgroup of G generated by S. If A, B C G, then (A, B) = gp((a,b)IaEA,bEB).IfA,B~G,then(A,B)~~ABand(A,B)(lG. G’ = (G, G) and G” = (G’)‘. The lower central series of G is defined inductively by: Gl = G G v+l = (Gv , (3 Evidently, G, = G’. Moreover, Gl 3 G, 3 ..
    [Show full text]
  • The Banach–Tarski Paradox and Amenability Lecture 25: Thompson's Group F
    The Banach{Tarski Paradox and Amenability Lecture 25: Thompson's Group F 30 October 2012 Thompson's groups F , T and V In 1965 Richard Thompson (a logician) defined 3 groups F ≤ T ≤ V Each is finitely presented and infinite, and Thompson used them to construct groups with unsolvable word problem. From the point of view of amenability F is the most important. The group F is not elementary amenable and does not have a subgroup which is free of rank ≥ 2. We do not know if F is amenable. The groups T and V are finitely presented infinite simple groups. The commutator subgroup [F ; F ] of F is finitely presented and ∼ 2 simple and F =[F ; F ] = Z . Definition of F We can define F to be the set of piecewise linear homeomorphisms f : [0; 1] ! [0; 1] which are differentiable except at finitely many a dyadic rationals (i.e. rationals 2n 2 (0; 1)), and such that on intervals of differentiability the derivatives are powers of 2. Examples See sketches of A and B. Lemma F is a subgroup of the group of all homeomorphisms [0; 1] ! [0; 1]. Proof. Let f 2 F have breakpoints 0 = x0 < x1 < ··· < xn = 1. Then f (x) = a1x on [0; x1] where a1 is a power of 2, so f (x1) = a1x1 is a dyadic rational. Hence f (x) = a2x + b2 on [x1; x2] where a2 is a power of 2 and b2 is dyadic rational. By induction f (x) = ai x + bi on [xi−1; xi ] with ai a power of 2 and bi a dyadic rational.
    [Show full text]
  • 0. Introduction to Categories
    0. Introduction to categories September 21, 2014 These are notes for Algebra 504-5-6. Basic category theory is very simple, and in principle one could read these notes right away|but only at the risk of being buried in an avalanche of abstraction. To get started, it's enough to understand the definition of \category" and \functor". Then return to the notes as necessary over the course of the year. The more examples you know, the easier it gets. Since this is an algebra course, the vast majority of the examples will be algebraic in nature. Occasionally I'll throw in some topological examples, but for the purposes of the course these can be omitted. 1 Definition and examples of a category A category C consists first of all of a class of objects Ob C and for each pair of objects X; Y a set of morphisms MorC(X; Y ). The elements of MorC(X; Y ) are denoted f : X−!Y , where f is the morphism, X is the source and Y is the target. For each triple of objects X; Y; Z there is a composition law MorC(X; Y ) × MorC(Y; Z)−!MorC(X; Z); denoted by (φ, ) 7! ◦ φ and subject to the following two conditions: (i) (associativity) h ◦ (g ◦ f) = (h ◦ g) ◦ f (ii) (identity) for every object X there is given an identity morphism IdX satisfying IdX ◦ f = f for all f : Y −!X and g ◦ IdX = g for all g : X−!Y . Note that the identity morphism IdX is the unique morphism satisfying (ii).
    [Show full text]
  • The Alternating Group
    The Alternating Group R. C. Daileda 1 The Parity of a Permutation Let n ≥ 2 be an integer. We have seen that any σ 2 Sn can be written as a product of transpositions, that is σ = τ1τ2 ··· τm; (1) where each τi is a transposition (2-cycle). Although such an expression is never unique, there is still an important invariant that can be extracted from (1), namely the parity of σ. Specifically, we will say that σ is even if there is an expression of the form (1) with m even, and that σ is odd if otherwise. Note that if σ is odd, then (1) holds only when m is odd. However, the converse is not immediately clear. That is, it is not a priori evident that a given permutation can't be expressed as both an even and an odd number of transpositions. Although it is true that this situation is, indeed, impossible, there is no simple, direct proof that this is the case. The easiest proofs involve an auxiliary quantity known as the sign of a permutation. The sign is uniquely determined for any given permutation by construction, and is easily related to the parity. This thereby shows that the latter is uniquely determined as well. The proof that we give below follows this general outline and is particularly straight- forward, given that the reader has an elementary knowledge of linear algebra. In particular, we assume familiarity with matrix multiplication and properties of the determinant. n We begin by letting e1; e2;:::; en 2 R denote the standard basis vectors, which are the columns of the n × n identity matrix: I = (e1 e2 ··· en): For σ 2 Sn, we define π(σ) = (eσ(1) eσ(2) ··· eσ(n)): Thus π(σ) is the matrix whose ith column is the σ(i)th column of the identity matrix.
    [Show full text]
  • MATH 436 Notes: Cyclic Groups and Invariant Subgroups
    MATH 436 Notes: Cyclic groups and Invariant Subgroups. Jonathan Pakianathan September 30, 2003 1 Cyclic Groups Now that we have enough basic tools, let us go back and study the structure of cyclic groups. Recall, these are exactly the groups G which can be generated by a single element x ∈ G. Before we begin we record an important example: Example 1.1. Given a group G and an element x ∈ G we may define a n homomorphism φx : Z → G via φx(n)= x . It is easy to see that Im(φx)=<x> is the cyclic subgroup generated by x. If one considers ker(φx), we have seen that it must be of the form dZ for some unique integer d ≥ 0. We define the order of x, denote o(x) by: d if d ≥ 1 o(x)= (∞ if d =0 Thus if x has infinite order then the elements in the set {xk|k ∈ Z} are all distinct while if x has order o(x) ≥ 1 then xk = e exactly when k is a multiple of o(x). Thus xm = xn whenever m ≡ n modulo o(x). By the First Isomorphism Theorem, <x>∼= Z/o(x)Z. Hence when x has finite order, | <x> | = o(x) and so in the case of a finite group, this order must divide the order of the group G by Lagrange’s Theorem. The homomorphisms above also give an important cautionary example! 1 Example 1.2 (Image subgroups are not necessarily normal). Let G = n Σ3 and a = (12) ∈ G. Then φa : Z → G defined by φa(n) = a has image Im(φa)=<a> which is not normal in G.
    [Show full text]
  • MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Basic Questions
    MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midterm Examination Model Solutions Basic Questions 1. Are the following multiplication tables groups? Justify your answers. a b c a c a c (a) b a b a c c a c This is not a group, since it does not have an identity element. It is also not assiciative: for example (ba)c = c, but b(ac) = a. a b c d e a a b c d e b b a e c d (b) c c d a e b d d e b a c e e c d b a This is not a group | it has 5 elements, and we know the only 5 element group is cyclic, which this is not, so it is not a group. [It is not associative: for example (bc)d = b, but b(cd) = d. It does have an identity and inverses for all elements.] a b c d a a b c d (c) b b a d c c c d a b d d c b a This is a group. It is the Klein 4 group Z2 × Z2. a is the identity, and all elements are their own inverses. 2. Which of the following are groups: (a) N = fn 2 Zjn > 0g with the operation a ∗ b given by addition without carrying, that is, write a and b (in decimal, including any leading zeros necessary) and in each position add the numbers modulo 10, so for example 2456 ∗ 824 = 2270. This is a group. 0 is the identity element.
    [Show full text]
  • A Geometric Study of Commutator Subgroups
    A Geometric Study of Commutator Subgroups Thesis by Dongping Zhuang In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2009 (Defended May 12, 2009) ii °c 2009 Dongping Zhuang All Rights Reserved iii To my parents iv Acknowledgments I am deeply indebted to my advisor Danny Calegari, whose guidance, stimulating suggestions and encour- agement help me during the time of research for and writing of this thesis. Thanks to Danny Calegari, Michael Aschbacher, Tom Graber and Matthew Day for graciously agreeing to serve on my Thesis Advisory Committee. Thanks to Koji Fujiwara, Jason Manning, Pierre Py and Matthew Day for inspiring discussions about the work contained in this thesis. Thanks to my fellow graduate students. Special thanks to Joel Louwsma for helping improve the writing of this thesis. Thanks to Nathan Dunfield, Daniel Groves, Mladen Bestvina, Kevin Wortman, Irine Peng, Francis Bonahon, Yi Ni and Jiajun Wang for illuminating mathematical conversations. Thanks to the Caltech Mathematics Department for providing an excellent environment for research and study. Thanks to the Department of Mathematical and Computing Sciences of Tokyo Institute of Technology; to Sadayoshi Kojima for hosting my stay in Japan; to Eiko Kin and Mitsuhiko Takasawa for many interesting mathematical discussions. Finally, I would like to give my thanks to my family and my friends whose patience and support enable me to complete this work. v Abstract Let G be a group and G0 its commutator subgroup. Commutator length (cl) and stable commutator length (scl) are naturally defined concepts for elements of G0.
    [Show full text]