DOCSLIB.ORG

The Applied Algebra Workbook

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

The Applied Algebra Workbook David Nacin William Paterson University March 22, 2018 2 Contents 1 Some Basic Results in Group Theory5 1.1 The Dihedral Groups.........................................5 1.1.1 Arbitrary Dihedral Group Questions............................6 1.1.2 The Group D3 ........................................8 1.1.3 The Group D4 ........................................ 11 1.1.4 The Group D5 ........................................ 14 1.1.5 The Group D6 ........................................ 17 1.2 The Symmetric Groups....................................... 21 1.2.1 The Group S3 ........................................ 23 1.2.2 The Group S4 ........................................ 26 1.2.3 The Group S5 ........................................ 30 1.2.4 Dihedral Groups as Subgroups of Symmetric Groups................... 34 1.3 Groups of Rotational Symmetries.................................. 37 1.3.1 The Tetrahedron....................................... 37 1.3.2 The Cube........................................... 43 1.3.3 The Dodecahedron...................................... 47 1.4 Group Actions............................................ 52 2 The Orbit-Stabilizer Theorem 59 2.1 Matrices................................................ 59 2.2 Necklaces............................................... 65 2.3 Platonic Solids............................................ 79 2.3.1 The Cube........................................... 79 2.3.2 The Tetrahedron....................................... 93 2.3.3 The Octahedron....................................... 101 2.3.4 The Dodecahedron...................................... 107 2.3.5 Additional Comments.................................... 109 2.4 Orbit Stabilizer Questions on Graphs................................ 111 2.4.1 Graphs and Labeled Graphs on Three Vertices...................... 111 2.4.2 Graphs and Labeled Graphs on Four Vertices....................... 113 2.5 The Proof of the Orbit Stabilizer Theorem............................. 119 3 4 CONTENTS 3 The Cauchy-Frobenius Lemma 125 3.1 Necklaces............................................... 125 3.2 Graphs................................................. 142 3.3 Platonic Solids............................................ 147 3.4 Matrices................................................ 168 3.5 The Proof of the Cauchy Frobenius Theorem........................... 176 4 P´olya Enumeration Questions 181 4.1 Necklaces............................................... 181 4.2 Graphs................................................. 187 4.3 Platonic Solids............................................ 191 4.4 Matrices................................................ 194 5 Irreducible Polynomials Over Finite Fields 201 5.1 Finding Irreducible Polynomials Over Z2 .............................. 202 5.2 Finding Irreducible Polynomials Over Z3 .............................. 206 5.3 Finding Irreducible Polynomials of Over Z5 ............................ 210 5.4 Finding Irreducible Polynomials of Over Z7 ............................ 215 5.5 Finding Irreducible Polynomials of Over Z11 ............................ 220 6 Construction of Finite Fields 223 6.1 Finite Fields of Order 2n....................................... 223 6.2 Finite Fields of Order 3n....................................... 226 6.3 Finite Fields of Order 5n....................................... 231 6.4 Finite Fields of Order 7n....................................... 235 6.5 Finite Fields of Order 11n....................................... 237 7 Linear Codes 243 7.1 Linear [n; k; d]-Codes and Duals................................... 243 7.2 Hadamard Codes........................................... 247 7.3 Automorphism Groups of Codes.................................. 252 8 Acknowledgments 257 Chapter 1 Some Basic Results in Group Theory 1.1 The Dihedral Groups There are many different ways one can define the dihedral groups. It's helpful to first look at them as actual reflections and rotations of some object. We can define these groups by flipping and rotating this object and then making a multiplication table out of the results. This becomes impractical for large values of n, but works very well up to about D5. For an example, we can start by taking a college textbook, or to be precise, a two-dimensional image of one. Picture it lying face up in some plane, and picture also a line straight through the middle of the book. There are many choices for such a line, and it actually doesn't matter which we choose, but it's probably easiest to imagine one parallel to two of the edges of the book. We start by defining two group elements. The element f in Dn corresponds to the motion of flipping the book over that axis, so it is now face down with the top of the book pointing the opposite direction as before. The element r corresponds to rotating the book clockwise 360=n degrees through a point directly in its center. For example, r2f is the motion we get from flipping and then rotating twice, and frfr2 is the motion we get from rotating twice, flipping, rotating, and then flipping again. We perform these operations from right to left in all cases in order to match the common notation for compositions used in Pre-Calculus and Calculus, where f ◦ g equals f(g(x)) and g ends up applied first. We have an equivalence relation on the set of all the possible motions that arise from combining these operations, considering two of these motions to be equal if they affect our object in the same way. This set of equivalence classes under composition forms the dihedral group. It helps to examine the relationship between f and r that arises because we set two of these motions to be equivalent if they leave the book in the same position. For example, in D4, if we rotate and flip, the book is in the same position as if we had flipped first and rotated three times, and thus fr = r3f: This is illustrated in figure 1.1 where we label a square book with the numbers one through four, and illustrate fr, rf and then r3f. The dotted line depicts the axis for reflection, which does not change as the book is rotated. Every 2 n−1 2 n−1 motion in Dn ends up equivalent to one of the following: e; r; r ; ··· ; r ; f; rf; r f; ··· ; r f and thus Dn contains 2n elements. It is important to keep in mind that the axis over which we flip the book does not rotate together with the object we are rotating. No matter which position the book is in, when we apply the flip, we always flip over the initial axis. It is also important to note that the elements in the group are the actual motions, and not the book itself. We use the positions to determine if two motions are equivalent, by seeing if the leave 5 6 CHAPTER 1. SOME BASIC RESULTS IN GROUP THEORY 3 Figure 1.1: Showing that fr 6= rf and fr = r f in D4 the book in the same place. We understand that this method of first defining the dihedral groups is not particularly rigorous. For 501 499 example, in D1000 it would be particularly difficult to see that fr is equal to r f: However, for small enough values of n one can get through most of the questions simply by taking a book and making these flips, and we feel it is nice to introduce these groups through an actual process that students can put their hands on. Another way to construct Dn is by looking at it as a subgroup of the symmetric group Sn. For even n this is the smallest group containing the permutation (1; 2; 3; ··· ; n) which represents r and (1; n)(2; n − 1) ··· (n=2; (n=2) + 1) which represents f. This will sometimes be useful when the numbers correspond to the physical corners and edges of some object. If we recall how to multiply in the symmetric group we can find 3 the relations for our group by simply taking products. For example we can see that rf = fr in D4 because (1; 2; 3; 4)(1; 4)(2; 3) and (1; 4)(2; 3)(1; 2; 3; 4)3 both equal (2; 4). There is an alternate construction for odd n where r is still (1; 2; 3; ··· ; n) but f is now (1; n)(2; n − 1) ··· ((n − 1)=2; (n + 1)=2): We will look into this construction more in the section on symmetric groups. n 2 For these problems it is enough to consider the group presented as Dn =< r; fjr = f = e; fr = rn−1f >. There is more than one form for this presentation, Note that there are some prettier ways to do this, but this form has some distinct advantages. Every element can be written in the form rif j where i 2 f0; 1; 2; ··· ; (n − 1)g and j 2 f0; 1g. Then we can quickly simplify any product simply by pushing every r to the right of an f past that f, turning it into a rn−1. 1.1.1 Arbitrary Dihedral Group Questions 1. Use the fact that fr = rn−1f to prove that frk = rn−kf. [Answer: Moving each r past our f shows frk = (rn−1)kf = rnk−kf = rn(k−1)+n−k = (rn)k−1rn−k = ek−1rn−k = rn−k: Note that this is equivalent to the statement (n − 1)k ≡ −k modulo n.] k 2. Prove that for any k ≥ 0, an element of the form r f is its own inverse in Dn. [Answer: We just need to show the square of any such element is the identity. (rnf)2 = (rnf)(rnf) = 1.1. THE DIHEDRAL GROUPS 7 2 2 rn(rn−1)nff = rn(rn(n−1))e = r(n+n −n) = r(n ) = (rn)n = en = e:] 3. Prove that for odd n, Dn has n elements of order two. [Answer: We have already shown that rkf has order two for 0 ≤ k < n. This gives us n elements of order two. We must show there are no more. The identity always has order one. We have to show that no element from r; r2; ··· ; rn−1 is its own inverses. Take any rk for 0 < k < n. Then (rk)2 = r2k. If this is the identity then n would have to divide 2k. As n is odd, it would also have to divide k. This cannot happen with 0 < k < n.] 4. Prove that for even n, Dn has n + 1 elements of order two. [Answer: We have already shown that rkf has order two for 0 ≤ k < n. The identity has order one. We must show that only one element from r; r2; ··· ; rn−1 has order two.
Recommended publications
  • Math 411 Midterm 2, Thursday 11/17/11, 7PM-8:30PM. Instructions: Exam Time Is 90 Mins

    Math 411 Midterm 2, Thursday 11/17/11, 7PM-8:30PM. Instructions: Exam Time Is 90 Mins

    Math 411 Midterm 2, Thursday 11/17/11, 7PM-8:30PM. Instructions: Exam time is 90 mins. There are 7 questions for a total of 75 points. Calculators, notes, and textbook are not allowed. Justify all your answers carefully. If you use a result proved in the textbook or class notes, state the result precisely. 1 Q1. (10 points) Let σ 2 S9 be the permutation 1 2 3 4 5 6 7 8 9 5 8 7 2 3 9 1 4 6 (a) (2 points) Write σ as a product of disjoint cycles. (b) (2 points) What is the order of σ? (c) (2 points) Write σ as a product of transpositions. (d) (4 points) Compute σ100. Q2. (8 points) (a) (3 points) Find an element of S5 of order 6 or prove that no such element exists. (b) (5 points) Find an element of S6 of order 7 or prove that no such element exists. Q3. (12 points) (a) (4 points) List all the elements of the alternating group A4. (b) (8 points) Find the left cosets of the subgroup H of A4 given by H = fe; (12)(34); (13)(24); (14)(23)g: Q4. (10 points) (a) (3 points) State Lagrange's theorem. (b) (7 points) Let p be a prime, p ≥ 3. Let Dp be the dihedral group of symmetries of a regular p-gon (a polygon with p sides of equal length). What are the possible orders of subgroups of Dp? Give an example in each case. 2 Q5. (13 points) (a) (3 points) State a theorem which describes all finitely generated abelian groups.
  • Generating the Mathieu Groups and Associated Steiner Systems

    Generating the Mathieu Groups and Associated Steiner Systems

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 112 (1993) 41-47 41 North-Holland Generating the Mathieu groups and associated Steiner systems Marston Conder Department of Mathematics and Statistics, University of Auckland, Private Bag 92019, Auckland, New Zealand Received 13 July 1989 Revised 3 May 1991 Abstract Conder, M., Generating the Mathieu groups and associated Steiner systems, Discrete Mathematics 112 (1993) 41-47. With the aid of two coset diagrams which are easy to remember, it is shown how pairs of generators may be obtained for each of the Mathieu groups M,,, MIz, Mz2, M,, and Mz4, and also how it is then possible to use these generators to construct blocks of the associated Steiner systems S(4,5,1 l), S(5,6,12), S(3,6,22), S(4,7,23) and S(5,8,24) respectively. 1. Introduction Suppose you land yourself in 24-dimensional space, and are faced with the problem of finding the best way to pack spheres in a box. As is well known, what you really need for this is the Leech Lattice, but alas: you forgot to bring along your Miracle Octad Generator. You need to construct the Leech Lattice from scratch. This may not be so easy! But all is not lost: if you can somehow remember how to define the Mathieu group Mz4, you might be able to produce the blocks of a Steiner system S(5,8,24), and the rest can follow. In this paper it is shown how two coset diagrams (which are easy to remember) can be used to obtain pairs of generators for all five of the Mathieu groups M,,, M12, M22> Mz3 and Mz4, and also how blocks may be constructed from these for each of the Steiner systems S(4,5,1 I), S(5,6,12), S(3,6,22), S(4,7,23) and S(5,8,24) respectively.
  • Classification of Finite Abelian Groups

    Classification of Finite Abelian Groups

    Math 317 C1 John Sullivan Spring 2003 Classification of Finite Abelian Groups (Notes based on an article by Navarro in the Amer. Math. Monthly, February 2003.) The fundamental theorem of finite abelian groups expresses any such group as a product of cyclic groups: Theorem. Suppose G is a finite abelian group. Then G is (in a unique way) a direct product of cyclic groups of order pk with p prime. Our first step will be a special case of Cauchy’s Theorem, which we will prove later for arbitrary groups: whenever p |G| then G has an element of order p. Theorem (Cauchy). If G is a finite group, and p |G| is a prime, then G has an element of order p (or, equivalently, a subgroup of order p). ∼ Proof when G is abelian. First note that if |G| is prime, then G = Zp and we are done. In general, we work by induction. If G has no nontrivial proper subgroups, it must be a prime cyclic group, the case we’ve already handled. So we can suppose there is a nontrivial subgroup H smaller than G. Either p |H| or p |G/H|. In the first case, by induction, H has an element of order p which is also order p in G so we’re done. In the second case, if ∼ g + H has order p in G/H then |g + H| |g|, so hgi = Zkp for some k, and then kg ∈ G has order p. Note that we write our abelian groups additively. Definition. Given a prime p, a p-group is a group in which every element has order pk for some k.
  • On Some Generation Methods of Finite Simple Groups

    On Some Generation Methods of Finite Simple Groups

    Introduction Preliminaries Special Kind of Generation of Finite Simple Groups The Bibliography On Some Generation Methods of Finite Simple Groups Ayoub B. M. Basheer Department of Mathematical Sciences, North-West University (Mafikeng), P Bag X2046, Mmabatho 2735, South Africa Groups St Andrews 2017 in Birmingham, School of Mathematics, University of Birmingham, United Kingdom 11th of August 2017 Ayoub Basheer, North-West University, South Africa Groups St Andrews 2017 Talk in Birmingham Introduction Preliminaries Special Kind of Generation of Finite Simple Groups The Bibliography Abstract In this talk we consider some methods of generating finite simple groups with the focus on ranks of classes, (p; q; r)-generation and spread (exact) of finite simple groups. We show some examples of results that were established by the author and his supervisor, Professor J. Moori on generations of some finite simple groups. Ayoub Basheer, North-West University, South Africa Groups St Andrews 2017 Talk in Birmingham Introduction Preliminaries Special Kind of Generation of Finite Simple Groups The Bibliography Introduction Generation of finite groups by suitable subsets is of great interest and has many applications to groups and their representations. For example, Di Martino and et al. [39] established a useful connection between generation of groups by conjugate elements and the existence of elements representable by almost cyclic matrices. Their motivation was to study irreducible projective representations of the sporadic simple groups. In view of applications, it is often important to exhibit generating pairs of some special kind, such as generators carrying a geometric meaning, generators of some prescribed order, generators that offer an economical presentation of the group.
  • Janko's Sporadic Simple Groups

    Janko's Sporadic Simple Groups

    Janko’s Sporadic Simple Groups: a bit of history Algebra, Geometry and Computation CARMA, Workshop on Mathematics and Computation Terry Gagen and Don Taylor The University of Sydney 20 June 2015 Fifty years ago: the discovery In January 1965, a surprising announcement was communicated to the international mathematical community. Zvonimir Janko, working as a Research Fellow at the Institute of Advanced Study within the Australian National University had constructed a new sporadic simple group. Before 1965 only five sporadic simple groups were known. They had been discovered almost exactly one hundred years prior (1861 and 1873) by Émile Mathieu but the proof of their simplicity was only obtained in 1900 by G. A. Miller. Finite simple groups: earliest examples É The cyclic groups Zp of prime order and the alternating groups Alt(n) of even permutations of n 5 items were the earliest simple groups to be studied (Gauss,≥ Euler, Abel, etc.) É Evariste Galois knew about PSL(2,p) and wrote about them in his letter to Chevalier in 1832 on the night before the duel. É Camille Jordan (Traité des substitutions et des équations algébriques,1870) wrote about linear groups defined over finite fields of prime order and determined their composition factors. The ‘groupes abéliens’ of Jordan are now called symplectic groups and his ‘groupes hypoabéliens’ are orthogonal groups in characteristic 2. É Émile Mathieu introduced the five groups M11, M12, M22, M23 and M24 in 1861 and 1873. The classical groups, G2 and E6 É In his PhD thesis Leonard Eugene Dickson extended Jordan’s work to linear groups over all finite fields and included the unitary groups.
  • Arxiv:Hep-Th/0510191V1 22 Oct 2005

    Arxiv:Hep-Th/0510191V1 22 Oct 2005

    Maximal Subgroups of the Coxeter Group W (H4) and Quaternions Mehmet Koca,∗ Muataz Al-Barwani,† and Shadia Al-Farsi‡ Department of Physics, College of Science, Sultan Qaboos University, PO Box 36, Al-Khod 123, Muscat, Sultanate of Oman Ramazan Ko¸c§ Department of Physics, Faculty of Engineering University of Gaziantep, 27310 Gaziantep, Turkey (Dated: September 8, 2018) The largest finite subgroup of O(4) is the noncrystallographic Coxeter group W (H4) of order 14400. Its derived subgroup is the largest finite subgroup W (H4)/Z2 of SO(4) of order 7200. Moreover, up to conjugacy, it has five non-normal maximal subgroups of orders 144, two 240, 400 and 576. Two groups [W (H2) × W (H2)]×Z4 and W (H3)×Z2 possess noncrystallographic structures with orders 400 and 240 respectively. The groups of orders 144, 240 and 576 are the extensions of the Weyl groups of the root systems of SU(3)×SU(3), SU(5) and SO(8) respectively. We represent the maximal subgroups of W (H4) with sets of quaternion pairs acting on the quaternionic root systems. PACS numbers: 02.20.Bb INTRODUCTION The noncrystallographic Coxeter group W (H4) of order 14400 generates some interests [1] for its relevance to the quasicrystallographic structures in condensed matter physics [2, 3] as well as its unique relation with the E8 gauge symmetry associated with the heterotic superstring theory [4]. The Coxeter group W (H4) [5] is the maximal finite subgroup of O(4), the finite subgroups of which have been classified by du Val [6] and by Conway and Smith [7].
  • Orders on Computable Torsion-Free Abelian Groups

    Orders on Computable Torsion-Free Abelian Groups

    Orders on Computable Torsion-Free Abelian Groups Asher M. Kach (Joint Work with Karen Lange and Reed Solomon) University of Chicago 12th Asian Logic Conference Victoria University of Wellington December 2011 Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 1 / 24 Outline 1 Classical Algebra Background 2 Computing a Basis 3 Computing an Order With A Basis Without A Basis 4 Open Questions Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 2 / 24 Torsion-Free Abelian Groups Remark Disclaimer: Hereout, the word group will always refer to a countable torsion-free abelian group. The words computable group will always refer to a (fixed) computable presentation. Definition A group G = (G : +; 0) is torsion-free if non-zero multiples of non-zero elements are non-zero, i.e., if (8x 2 G)(8n 2 !)[x 6= 0 ^ n 6= 0 =) nx 6= 0] : Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 3 / 24 Rank Theorem A countable abelian group is torsion-free if and only if it is a subgroup ! of Q . Definition The rank of a countable torsion-free abelian group G is the least κ cardinal κ such that G is a subgroup of Q . Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 4 / 24 Example The subgroup H of Q ⊕ Q (viewed as having generators b1 and b2) b1+b2 generated by b1, b2, and 2 b1+b2 So elements of H look like β1b1 + β2b2 + α 2 for β1; β2; α 2 Z.
  • 14.2 Covers of Symmetric and Alternating Groups

    14.2 Covers of Symmetric and Alternating Groups

    Remark 206 In terms of Galois cohomology, there an exact sequence of alge- braic groups (over an algebrically closed field) 1 → GL1 → ΓV → OV → 1 We do not necessarily get an exact sequence when taking values in some subfield. If 1 → A → B → C → 1 is exact, 1 → A(K) → B(K) → C(K) is exact, but the map on the right need not be surjective. Instead what we get is 1 → H0(Gal(K¯ /K), A) → H0(Gal(K¯ /K), B) → H0(Gal(K¯ /K), C) → → H1(Gal(K¯ /K), A) → ··· 1 1 It turns out that H (Gal(K¯ /K), GL1) = 1. However, H (Gal(K¯ /K), ±1) = K×/(K×)2. So from 1 → GL1 → ΓV → OV → 1 we get × 1 1 → K → ΓV (K) → OV (K) → 1 = H (Gal(K¯ /K), GL1) However, taking 1 → µ2 → SpinV → SOV → 1 we get N × × 2 1 ¯ 1 → ±1 → SpinV (K) → SOV (K) −→ K /(K ) = H (K/K, µ2) so the non-surjectivity of N is some kind of higher Galois cohomology. Warning 207 SpinV → SOV is onto as a map of ALGEBRAIC GROUPS, but SpinV (K) → SOV (K) need NOT be onto. Example 208 Take O3(R) =∼ SO3(R)×{±1} as 3 is odd (in general O2n+1(R) =∼ SO2n+1(R) × {±1}). So we have a sequence 1 → ±1 → Spin3(R) → SO3(R) → 1. 0 × Notice that Spin3(R) ⊆ C3 (R) =∼ H, so Spin3(R) ⊆ H , and in fact we saw that it is S3. 14.2 Covers of symmetric and alternating groups The symmetric group on n letter can be embedded in the obvious way in On(R) as permutations of coordinates.
  • 324 COHOMOLOGY of DIHEDRAL GROUPS of ORDER 2P Let D Be A

    324 COHOMOLOGY of DIHEDRAL GROUPS of ORDER 2P Let D Be A

    324 PHYLLIS GRAHAM [March 2. P. M. Cohn, Unique factorization in noncommutative power series rings, Proc. Cambridge Philos. Soc 58 (1962), 452-464. 3. Y. Kawada, Cohomology of group extensions, J. Fac. Sci. Univ. Tokyo Sect. I 9 (1963), 417-431. 4. J.-P. Serre, Cohomologie Galoisienne, Lecture Notes in Mathematics No. 5, Springer, Berlin, 1964, 5. , Corps locaux et isogênies, Séminaire Bourbaki, 1958/1959, Exposé 185 6. , Corps locaux, Hermann, Paris, 1962. UNIVERSITY OF MICHIGAN COHOMOLOGY OF DIHEDRAL GROUPS OF ORDER 2p BY PHYLLIS GRAHAM Communicated by G. Whaples, November 29, 1965 Let D be a dihedral group of order 2p, where p is an odd prime. D is generated by the elements a and ]3 with the relations ap~fi2 = 1 and (ial3 = or1. Let A be the subgroup of D generated by a, and let p-1 Aoy Ai, • • • , Ap-\ be the subgroups generated by j8, aft, • * • , a /3, respectively. Let M be any D-module. Then the cohomology groups n n H (A0, M) and H (Aif M), i=l, 2, • • • , p — 1 are isomorphic for 1 l every integer n, so the eight groups H~ (Df Af), H°(D, M), H (D, jfef), 2 1 Q # (A Af), ff" ^. M), H (A, M), H-Wo, M), and H»(A0, M) deter­ mine all cohomology groups of M with respect to D and to all of its subgroups. We have found what values this array takes on as M runs through all finitely generated -D-modules. All possibilities for the first four members of this array are deter­ mined as a special case of the results of Yang [4].
  • Dihedral Groups Ii

    Dihedral Groups Ii

    DIHEDRAL GROUPS II KEITH CONRAD We will characterize dihedral groups in terms of generators and relations, and describe the subgroups of Dn, including the normal subgroups. We will also introduce an infinite group that resembles the dihedral groups and has all of them as quotient groups. 1. Abstract characterization of Dn −1 −1 The group Dn has two generators r and s with orders n and 2 such that srs = r . We will show every group with a pair of generators having properties similar to r and s admits a homomorphism onto it from Dn, and is isomorphic to Dn if it has the same size as Dn. Theorem 1.1. Let G be generated by elements x and y where xn = 1 for some n ≥ 3, 2 −1 −1 y = 1, and yxy = x . There is a surjective homomorphism Dn ! G, and if G has order 2n then this homomorphism is an isomorphism. The hypotheses xn = 1 and y2 = 1 do not mean x has order n and y has order 2, but only that their orders divide n and divide 2. For instance, the trivial group has the form hx; yi where xn = 1, y2 = 1, and yxy−1 = x−1 (take x and y to be the identity). Proof. The equation yxy−1 = x−1 implies yxjy−1 = x−j for all j 2 Z (raise both sides to the jth power). Since y2 = 1, we have for all k 2 Z k ykxjy−k = x(−1) j by considering even and odd k separately. Thus k (1.1) ykxj = x(−1) jyk: This shows each product of x's and y's can have all the x's brought to the left and all the y's brought to the right.
  • Countable Torsion-Free Abelian Groups

    Countable Torsion-Free Abelian Groups

    COUNTABLE TORSION-FREE ABELIAN GROUPS By M. O'N. CAMPBELL [Received 27 January 1959.—Read 19 February 1959] Introduction IN this paper we present a new classification of the countable torsion-free abelian groups. Since any abelian group 0 may be regarded as an extension of a periodic group (namely the torsion part of G) by a torsion-free group, and since there exists the well-known complete classification of the count- able periodic abelian groups due to Ulm, it is clear that the classification problem studied here is of great interest. By a group we shall always mean an abelian group, and the additive notation will be employed throughout. It is well known that all the maximal linearly independent sets of elements of a given group are car- dinally equivalent; the corresponding cardinal number is the rank of the group. Derry (2) and Mal'cev (4) have studied the torsion-free groups of finite rank only, and have given classifications of these groups in terms of certain equivalence classes of systems of matrices with £>-adic elements. Szekeres (5) attempted to abolish the finiteness restriction on rank; he extracted certain invariants, but did not derive a complete classification. By a basis of a torsion-free group 0 we mean any maximal linearly independent subset of G. A subgroup generated by a basis of G will be called a basal subgroup or described as basal in G. Thus U is a basal sub- group of G if and only if (i) U is free abelian, and (ii) the factor group G/U is periodic.
  • Quasi P Or Not Quasi P? That Is the Question

    Quasi P Or Not Quasi P? That Is the Question

    Rose-Hulman Undergraduate Mathematics Journal Volume 3 Issue 2 Article 2 Quasi p or not Quasi p? That is the Question Ben Harwood Northern Kentucky University, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Harwood, Ben (2002) "Quasi p or not Quasi p? That is the Question," Rose-Hulman Undergraduate Mathematics Journal: Vol. 3 : Iss. 2 , Article 2. Available at: https://scholar.rose-hulman.edu/rhumj/vol3/iss2/2 Quasi p- or not quasi p-? That is the Question.* By Ben Harwood Department of Mathematics and Computer Science Northern Kentucky University Highland Heights, KY 41099 e-mail: [email protected] Section Zero: Introduction The question might not be as profound as Shakespeare’s, but nevertheless, it is interesting. Because few people seem to be aware of quasi p-groups, we will begin with a bit of history and a definition; and then we will determine for each group of order less than 24 (and a few others) whether the group is a quasi p-group for some prime p or not. This paper is a prequel to [Hwd]. In [Hwd] we prove that (Z3 £Z3)oZ2 and Z5 o Z4 are quasi 2-groups. Those proofs now form a portion of Proposition (12.1) It should also be noted that [Hwd] may also be found in this journal. Section One: Why should we be interested in quasi p-groups? In a 1957 paper titled Coverings of algebraic curves [Abh2], Abhyankar conjectured that the algebraic fundamental group of the affine line over an algebraically closed field k of prime characteristic p is the set of quasi p-groups, where by the algebraic fundamental group of the affine line he meant the family of all Galois groups Gal(L=k(X)) as L varies over all finite normal extensions of k(X) the function field of the affine line such that no point of the line is ramified in L, and where by a quasi p-group he meant a finite group that is generated by all of its p-Sylow subgroups.