DOCSLIB.ORG

Math 375 Week 5

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math 375 Week 5 Math 375 Week 5 5.1 Symmetric Groups The Group S X Now let X be any non-empty set. Let S = fj : X ! X and f is b oth surjective and injective g: X Notice that if ; 2 S , then : X ! X and is surjective and injec- X tiveby previous arguments. That is, the comp osition 2 S .Thus X comp osition is a binary op eration on S . X THEOREM 1 S is a group under comp osition. S is called the Symmteric Group X X on X . PROOF i We just checked closure. ii Asso ciativity: mentioned last time see text Chapter 0. iii The identityinS is i b ecause X x= X X i x = x, that is, i = . Similarly i = . iv Evidently X X X 1 the inverse of is whichwe know exists b ecause is injective and 1 1 surjective and x= x = i x and x= x = i x. X X This last example shows just how general the group concept is. Sym- metric groups are extraordinarily imp ortant. Arthur Cayley as in Cay- ley table showed that every group is the subgroup of some symmetric group. So if you understand symmetric groups completely, then you un- derstand all groups! We can examine S for any set X .For example X if X = R, then examples of of elements in S are i , f : R ! R by R R a ! a +1, g : R ! R by a ! a=2, and so on. It is clear that S is R in nite. The individual elements of S are often called p ermutations of X the elements of X . This will b e the next big topic wecover. 1 2 Math 375 The Symmetric Group on n Elements: S n To make matters simpler, we will study symmetric groups of nite sets. For example, if X is a set of n elements, then wemayaswell lab el the elements of X as f1; 2;:::;ng.We usually denote the symmetric group on n elements by S . n Nowany elementorp ermutation in S is an injective and sur- n jective function from the set of the rst n integers to itself. It merely shues these elements around. Consequently it can b e represented as atworow matrix in which the rst row represents the input and the second represents the corresp onding output. EXAMPLE 1 List all the elements of S . 3 1 2 3 1 2 3 1 2 3 2 = ; = ; = 1 2 3 2 3 1 3 1 2 1 2 3 1 2 3 1 2 3 2 = ; = ; = 1 3 2 3 2 1 2 1 3 It should b e clear what the identity element is. Howdowe get the inverse of any element? Merely exchange rows, and re-order. How do we take the pro duct or comp osition of two such elements in S ? n In comp osing functions always remember to work from the right to the left whichisbackwards from what we read. This is a hold over from comp osition of functions, which is what we implicitly are doing here. 1 2 3 1 2 3 1 2 3 = 3 2 1 2 1 3 2 3 1 Notice that order matters: 1 2 3 1 2 3 1 2 3 = 2 1 3 3 2 1 3 1 2 Clearly S is ab elian, since it consits of only the identity element. 1 We'll see that S is ab elian in fact it's cyclic since it has only two 2 elements. However, wehave seen that S is not ab elian and in general: 3 THEOREM 2 If n 3 then S is non-ab elian. n 5.1 Symmetric Groups 3 PROOF Let and b e the following two elements in S : n 1 2 3 4 ::: n 1 2 3 4 ::: n = = : 3 2 1 4 ::: n 2 1 3 4 ::: n Then 1 2 3 4 ::: n 1 2 3 4 ::: n = = : 3 1 2 4 ::: n 2 3 1 4 ::: n EXAMPLE 2 Find the following inverse: 1 1 2 3 1 2 3 = 2 3 1 3 1 2 THEOREM 3 jS j = n!: n PROOF jS j is just the numberofways the integers 1 through n can arranged. n In other words in how many di erentways can we ll in the blanks: 1 2 ::: n ::: Well wehave n choices for the rst entry, and then n 1choices for the next entry, and so on yielding a total of n n 1 1= n! total choices. For example jS j =3!=6aswehave already seen. jS j = 2!=2. 3 2 If we write out the Cayley table for S we see that it is ab elian. Note 2 that the square of every element is the identity. 1 2 1 2 1 2 2 1 1 2 1 2 1 2 1 2 1 2 2 1 1 2 1 2 1 2 2 1 2 1 1 2 EXAMPLE 3 Compare the symmetries of an equilateral triangle to S . Write each 3 elementinD as a p ermutation. 3 EXAMPLE 4 Compare D to S . Are they the same group? 4 4 4 Math 375 Cycle Notation It will provemuch more convenient to use a di ernttyp e of notation to denote the individual p ermutations of S . This cycle notation is useful n b ecause it reveals the structure of individual elements in the group, much as p owers of a generator indicate the nature of a cyclic group. Here's howitworks. Consider the following elements of S : 5 1 2 3 4 5 1 2 3 4 5 = = 2 3 1 5 4 4 3 5 1 2 Notice that breaks quite naturally into two parts: 1 ! 2 ! 3 ! 1 and 4 ! 5 ! 4 Each of these pieces is called a cycle. The rst is a three-cycle b ecause it contains three di erent elements, the second is a two-cycle. Eachof these cycles can b e represented more compactly without the arrows: 1; 2; 3 and 4; 5 Let's lo ok at . This time the cycles are: 1 ! 4 ! 1 and 2 ! 3 ! 5 ! 2 or 1; 4 and 2; 3; 5 When using cycles we adopt the convention that if an elementis missing from the cycle, then it is mapp ed to itself. EXAMPLE 5 What p ermutation do es the cycle 2; 4; 3 representin S ? 6 1 2 3 4 5 6 2; 4; 3 = 1 4 2 3 5 6 EXAMPLE 6 Let =1; 3; 62; 4; 5 and =1; 2; 34; 5 in S . Then = 6 4; 2; 6; 153 = 4; 2; 6; 1. Do some more. EXAMPLE 7 What is the inverse of a single cycle: =4; 2; 3; 5? This is easy to sp ot in matrix form. If 1 2 3 4 5 =4; 2; 3; 5 = ; 1 3 5 2 4 then 1 2 3 4 5 1 = =5; 3; 2; 4: 1 4 2 5 3 More generally: 5.1 Symmetric Groups 5 THEOREM 4 The inverse of a cycle is the cycle in inverse reverse order. Further the inverse of a pro duct of cycles is the pro duct of the inverse cycles in reverse order. REMARK The second part of the theorem is the usual statement ab out inverses of pro ducts b eing the pro duct of the inverses in reverse order. DEFINITION 5 Two cycles are called disjoint if they contain no term in common. For example 1; 2; 6 and 3; 4 are disjoint but 1; 2; 4 and 3; 4; 5 are not. One of the basic facts ab out p ermutations is that THEOREM 6 Any p ermutation can b e written as a pro duct of disjoint cycles. The details of the pro of are presented in the text, but note that this is howwe rst got started on lo oking at cycles. Given an individual example of a p ermutation it is easy to see how to split it into disjoint cycles. Consider 1 2 3 4 5 6 7 8 9 : 5 9 3 8 6 1 4 7 2 We simply start the rst cycle with 1, continue backuntil we get 1. Then start the second cycle with the smallest remaining unused num- ber until we get back to it and so on. In this case the cycles are: 1; 5; 62; 934; 8; 7: THEOREM 7 Disjoint cycles commute. PROOF Let the disjoint cycles b e =a ;a ;:::;a and =b ;b ;:::; , 1 2 m 1 2 n where and havenoentries in common. Let the full set S b e given by S = fa ;a ;:::;a ;b ;b ;:::; ;c ;:::;c g: 1 2 m 1 2 n 1 k Wewant to prove that = .To do this, wemust show that a = x for all x 2 S . There are three cases: x is either an a, b,or c. Supp ose that x = a . Then i x= a = a = a = a : i i i+1 mo d m i+1 mo d m while x= a = a = a = a : i i i i+1 mo d m Hence the p ermutations agree on the a's. A similar argumentworks for the b's. The c's are the easiest of all. If x = c , then i x= c = c = c = c i i i i while x = c = c = c = c : i i i i Again the p ermutations are the same. 6 Math 375 5.2 More ab out S n The Order of a k -cycle EXAMPLE 8 Consider the following elements of S : =2; 3 and =1; 2; 4, and 5 =1; 3; 4; 2 Find the orders of each of these elements.
Load more

Recommended publications
  • 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.
    [Show full text]
  • Complex Algebras of Semigroups Pamela Jo Reich Iowa State University
    Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1996 Complex algebras of semigroups Pamela Jo Reich Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Reich, Pamela Jo, "Complex algebras of semigroups " (1996). Retrospective Theses and Dissertations. 11765. https://lib.dr.iastate.edu/rtd/11765 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfihn master. UMI fibns the text du-ectly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproductioii is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and unproper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., m^s, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overiaps.
    [Show full text]
  • Group Isomorphisms MME 529 Worksheet for May 23, 2017 William J
    Group Isomorphisms MME 529 Worksheet for May 23, 2017 William J. Martin, WPI Goal: Illustrate the power of abstraction by seeing how groups arising in different contexts are really the same. There are many different kinds of groups, arising in a dizzying variety of contexts. Even on this worksheet, there are too many groups for any one of us to absorb. But, with different teams exploring different examples, we should { as a class { discover some justification for the study of groups in the abstract. The Integers Modulo n: With John Goulet, you explored the additive structure of Zn. Write down the addition table for Z5 and Z6. These groups are called cyclic groups: they are generated by a single element, the element 1, in this case. That means that every element can be found by adding 1 to itself an appropriate number of times. The Group of Units Modulo n: Now when we look at Zn using multiplication as our operation, we no longer have a group. (Why not?) The group U(n) = fa 2 Zn j gcd(a; n) = 1g ∗ is sometimes written Zn and is called the group of units modulo n. An element in a number system (or ring) is a \unit" if it has a multiplicative inverse. Write down the multiplication tables for U(6), U(7), U(8) and U(12). The Group of Rotations of a Regular n-Gon: Imagine a regular polygon with n sides centered at the origin O. Let e denote the identity transformation, which leaves the poly- gon entirely fixed and let a denote a rotation about O in the counterclockwise direction by exactly 360=n degrees (2π=n radians).
    [Show full text]
  • 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.
    [Show full text]
  • 4. Groups of Permutations 1
    4. Groups of permutations 1 4. Groups of permutations Consider a set of n distinguishable objects, fB1;B2;B3;:::;Bng. These may be arranged in n! different ways, called permutations of the set. Permu- tations can also be thought of as transformations of a given ordering of the set into other orderings. A convenient notation for specifying a given permutation operation is 1 2 3 : : : n ! , a1 a2 a3 : : : an where the numbers fa1; a2; a3; : : : ; ang are the numbers f1; 2; 3; : : : ; ng in some order. This operation can be interpreted in two different ways, as follows. Interpretation 1: The object in position 1 in the initial ordering is moved to position a1, the object in position 2 to position a2,. , the object in position n to position an. In this interpretation, the numbers in the two rows of the permutation symbol refer to the positions of objects in the ordered set. Interpretation 2: The object labeled 1 is replaced by the object labeled a1, the object labeled 2 by the object labeled a2,. , the object labeled n by the object labeled an. In this interpretation, the numbers in the two rows of the permutation symbol refer to the labels of the objects in the ABCD ! set. The labels need not be numerical { for instance, DCAB is a well-defined permutation which changes BDCA, for example, into CBAD. Either of these interpretations is acceptable, but one interpretation must be used consistently in any application. The particular application may dictate which is the appropriate interpretation to use. Note that, in either interpre- tation, the order of the columns in the permutation symbol is irrelevant { the columns may be written in any order without affecting the result, provided each column is kept intact.
    [Show full text]
  • 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.
    [Show full text]
  • PERMUTATION GROUPS (16W5087)
    PERMUTATION GROUPS (16w5087) Cheryl E. Praeger (University of Western Australia), Katrin Tent (University of Munster),¨ Donna Testerman (Ecole Polytechnique Fed´ erale´ de Lausanne), George Willis (University of Newcastle) November 13 - November 18, 2016 1 Overview of the Field Permutation groups are a mathematical approach to analysing structures by studying the rearrangements of the elements of the structure that preserve it. Finite permutation groups are primarily understood through combinatorial methods, while concepts from logic and topology come to the fore when studying infinite per- mutation groups. These two branches of permutation group theory are not completely independent however because techniques from algebra and geometry may be applied to both, and ideas transfer from one branch to the other. Our workshop brought together researchers on both finite and infinite permutation groups; the participants shared techniques and expertise and explored new avenues of study. The permutation groups act on discrete structures such as graphs or incidence geometries comprising points and lines, and many of the same intuitions apply whether the structure is finite or infinite. Matrix groups are also an important source of both finite and infinite permutation groups, and the constructions used to produce the geometries on which they act in each case are similar. Counting arguments are important when studying finite permutation groups but cannot be used to the same effect with infinite permutation groups. Progress comes instead by using techniques from model theory and descriptive set theory. Another approach to infinite permutation groups is through topology and approx- imation. In this approach, the infinite permutation group is locally the limit of finite permutation groups and results from the finite theory may be brought to bear.
    [Show full text]
  • Binding Groups, Permutation Groups and Modules of Finite Morley Rank Alexandre Borovik, Adrien Deloro
    Binding groups, permutation groups and modules of finite Morley rank Alexandre Borovik, Adrien Deloro To cite this version: Alexandre Borovik, Adrien Deloro. Binding groups, permutation groups and modules of finite Morley rank. 2019. hal-02281140 HAL Id: hal-02281140 https://hal.archives-ouvertes.fr/hal-02281140 Preprint submitted on 9 Sep 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. BINDING GROUPS, PERMUTATION GROUPS AND MODULES OF FINITE MORLEY RANK ALEXANDRE BOROVIK AND ADRIEN DELORO If one has (or if many people have) spent decades classifying certain objects, one is apt to forget just why one started the project in the first place. Predictions of the death of group theory in 1980 were the pronouncements of just such amne- siacs. [68, p. 4] Contents 1. Introduction and background 2 1.1. The classification programme 2 1.2. Concrete groups of finite Morley rank 3 2. Binding groups and bases 4 2.1. Binding groups 4 2.2. Bases and parametrisation 5 2.3. Sharp bounds for bases 6 3. Permutation groups of finite Morley rank 7 3.1. Highly generically multiply transitive groups 7 3.2.
    [Show full text]
  • Representations of the Infinite Symmetric Group
    Representations of the infinite symmetric group Alexei Borodin1 Grigori Olshanski2 Version of February 26, 2016 1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA Institute for Information Transmission Problems of the Russian Academy of Sciences, Moscow, Russia 2 Institute for Information Transmission Problems of the Russian Academy of Sciences, Moscow, Russia National Research University Higher School of Economics, Moscow, Russia Contents 0 Introduction 4 I Symmetric functions and Thoma's theorem 12 1 Symmetric group representations 12 Exercises . 19 2 Theory of symmetric functions 20 Exercises . 31 3 Coherent systems on the Young graph 37 The infinite symmetric group and the Young graph . 37 Coherent systems . 39 The Thoma simplex . 41 1 CONTENTS 2 Integral representation of coherent systems and characters . 44 Exercises . 46 4 Extreme characters and Thoma's theorem 47 Thoma's theorem . 47 Multiplicativity . 49 Exercises . 52 5 Pascal graph and de Finetti's theorem 55 Exercises . 60 6 Relative dimension in Y 61 Relative dimension and shifted Schur polynomials . 61 The algebra of shifted symmetric functions . 65 Modified Frobenius coordinates . 66 The embedding Yn ! Ω and asymptotic bounds . 68 Integral representation of coherent systems: proof . 71 The Vershik{Kerov theorem . 74 Exercises . 76 7 Boundaries and Gibbs measures on paths 82 The category B ............................. 82 Projective chains . 83 Graded graphs . 86 Gibbs measures . 88 Examples of path spaces for branching graphs . 91 The Martin boundary and Vershik{Kerov's ergodic theorem . 92 Exercises . 94 II Unitary representations 98 8 Preliminaries and Gelfand pairs 98 Exercises . 108 9 Spherical type representations 111 10 Realization of spherical representations 118 Exercises .
    [Show full text]
  • Arxiv:2001.06557V1 [Math.GR] 17 Jan 2020 Magic Cayley-Sudoku Tables
    Magic Cayley-Sudoku Tables∗ Rosanna Mersereau Michael B. Ward Columbus, OH Western Oregon University 1 Introduction Inspired by the popularity of sudoku puzzles along with the well-known fact that the body of the Cayley table1 of any finite group already has 2/3 of the properties of a sudoku table in that each element appears exactly once in each row and in each column, Carmichael, Schloeman, and Ward [1] in- troduced Cayley-sudoku tables. A Cayley-sudoku table of a finite group G is a Cayley table for G the body of which is partitioned into uniformly sized rectangular blocks, in such a way that each group element appears exactly once in each block. For example, Table 1 is a Cayley-sudoku ta- ble for Z9 := {0, 1, 2, 3, 4, 5, 6, 7, 8} under addition mod 9 and Table 3 is a Cayley-sudoku table for Z3 × Z3 where the operation is componentwise ad- dition mod 3 (and ordered pairs (a, b) are abbreviated ab). In each case, we see that we have a Cayley table of the group partitioned into 3 × 3 blocks that contain each group element exactly once. Lorch and Weld [3] defined a modular magic sudoku table as an ordinary sudoku table (with 0 in place of the usual 9) in which the row, column, arXiv:2001.06557v1 [math.GR] 17 Jan 2020 diagonal, and antidiagonal sums in each 3 × 3 block in the table are zero mod 9. “Magic” refers, of course, to magic Latin squares which have a rich history dating to ancient times.
    [Show full text]
  • 1.6 Cyclic Subgroups
    1.6. CYCLIC SUBGROUPS 19 1.6 Cyclic Subgroups Recall: cyclic subgroup, cyclic group, generator. Def 1.68. Let G be a group and a ∈ G. If the cyclic subgroup hai is finite, then the order of a is |hai|. Otherwise, a is of infinite order. 1.6.1 Elementary Properties Thm 1.69. Every cyclic group is abelian. Thm 1.70. If m ∈ Z+ and n ∈ Z, then there exist unique q, r ∈ Z such that n = mq + r and 0 ≤ r ≤ m. n In fact, q = b m c and r = n − mq. Here bxc denotes the maximal integer no more than x. Ex 1.71 (Ex 6.4, Ex 6.5, p60). 1. Find the quotient q and the remainder r when n = 38 is divided by m = 7. 2. Find the quotient q and the remainder r when n = −38 is divided by m = 7. Thm 1.72 (Important). A subgroup of a cyclic group is cyclic. Proof. (refer to the book) Ex 1.73. The subgroups of hZ, +i are precisely hnZ, +i for n ∈ Z. Def 1.74. Let r, s ∈ Z. The greatest common divisor (gcd) of r and s is the largest positive integer d that divides both r and s. Written as d = gcd(r, s). In fact, d is the positive generator of the following cyclic subgroup of Z: hdi = {nr + ms | n, m ∈ Z}. So d is the smallest positive integer that can be written as nr + ms for some n, m ∈ Z. Ex 1.75. gcd(36, 63) = 9, gcd(36, 49) = 1.
    [Show full text]
  • And Free Cyclic Group Actions on Homotopy Spheres
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 220, 1976 DECOMPOSABILITYOF HOMOTOPYLENS SPACES ANDFREE CYCLICGROUP ACTIONS ON HOMOTOPYSPHERES BY KAI WANG ABSTRACT. Let p be a linear Zn action on C and let p also denote the induced Z„ action on S2p~l x D2q, D2p x S2q~l and S2p~l x S2q~l " 1m_1 where p = [m/2] and q = m —p. A free differentiable Zn action (£ , ju) on a homotopy sphere is p-decomposable if there is an equivariant diffeomor- phism <t>of (S2p~l x S2q~l, p) such that (S2m_1, ju) is equivalent to (£(*), ¿(*)) where S(*) = S2p_1 x D2q U^, D2p x S2q~l and A(<P) is a uniquely determined action on S(*) such that i4(*)IS p~l XD q = p and A(Q)\D p X S = p. A homotopy lens space is p-decomposable if it is the orbit space of a p-decomposable free Zn action on a homotopy sphere. In this paper, we will study the decomposabilities of homotopy lens spaces. We will also prove that for each lens space L , there exist infinitely many inequivalent free Zn actions on S m such that the orbit spaces are simple homotopy equiva- lent to L 0. Introduction. Let A be the antipodal map and let $ be an equivariant diffeomorphism of (Sp x Sp, A) where A(x, y) = (-x, -y). Then there is a uniquely determined free involution A($) on 2(4>) where 2(4») = Sp x Dp+1 U<¡,Dp+l x Sp such that the inclusions S" x Dp+l —+ 2(d>), Dp+1 x Sp —*■2(4>) are equi- variant.
    [Show full text]