On Generic Polynomials for Cyclic Groups
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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. -
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. -
On Generic Polynomials for Dihedral Groups
ON GENERIC POLYNOMIALS FOR DIHEDRAL GROUPS ARNE LEDET Abstract. We provide an explicit method for constructing generic polynomials for dihedral groups of degree divisible by four over fields containing the appropriate cosines. 1. Introduction Given a field K and a finite group G, it is natural to ask what a Galois extension over K with Galois group G looks like. One way of formulating an answer is by means of generic polynomials: Definition. A monic separable polynomial P (t; X) 2 K(t)[X], where t = (t1; : : : ; tn) are indeterminates, is generic for G over K, if it satsifies the following conditions: (a) Gal(P (t; X)=K(t)) ' G; and (b) whenever M=L is a Galois extension with Galois group G and L ⊇ K, there exists a1; : : : ; an 2 L) such that M is the splitting field over L of the specialised polynomial P (a1; : : : ; an; X) 2 L[X]. The indeterminates t are the parameters. Thus, if P (t; X) is generic for G over K, every G-extension of fields containing K `looks just like' the splitting field of P (t; X) itself over K(t). This concept of a generic polynomial was shown by Kemper [Ke2] to be equivalent (over infinite fields) to the concept of a generic extension, as introduced by Saltman in [Sa]. For examples and further references, we refer to [JL&Y]. The inspiration for this paper came from [H&M], in which Hashimoto and Miyake describe a one-parameter generic polynomial for the dihe- dral group Dn of degree n (and order 2n), provided that n is odd, that char K - n, and that K contains the nth cosines, i.e., ζ + 1/ζ 2 K for a primitive nth root of unity ζ. -
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. -
A STUDY on the ALGEBRAIC STRUCTURE of SL 2(Zpz)
A STUDY ON THE ALGEBRAIC STRUCTURE OF SL2 Z pZ ( ~ ) A Thesis Presented to The Honors Tutorial College Ohio University In Partial Fulfillment of the Requirements for Graduation from the Honors Tutorial College with the degree of Bachelor of Science in Mathematics by Evan North April 2015 Contents 1 Introduction 1 2 Background 5 2.1 Group Theory . 5 2.2 Linear Algebra . 14 2.3 Matrix Group SL2 R Over a Ring . 22 ( ) 3 Conjugacy Classes of Matrix Groups 26 3.1 Order of the Matrix Groups . 26 3.2 Conjugacy Classes of GL2 Fp ....................... 28 3.2.1 Linear Case . .( . .) . 29 3.2.2 First Quadratic Case . 29 3.2.3 Second Quadratic Case . 30 3.2.4 Third Quadratic Case . 31 3.2.5 Classes in SL2 Fp ......................... 33 3.3 Splitting of Classes of(SL)2 Fp ....................... 35 3.4 Results of SL2 Fp ..............................( ) 40 ( ) 2 4 Toward Lifting to SL2 Z p Z 41 4.1 Reduction mod p ...............................( ~ ) 42 4.2 Exploring the Kernel . 43 i 4.3 Generalizing to SL2 Z p Z ........................ 46 ( ~ ) 5 Closing Remarks 48 5.1 Future Work . 48 5.2 Conclusion . 48 1 Introduction Symmetries are one of the most widely-known examples of pure mathematics. Symmetry is when an object can be rotated, flipped, or otherwise transformed in such a way that its appearance remains the same. Basic geometric figures should create familiar examples, take for instance the triangle. Figure 1: The symmetries of a triangle: 3 reflections, 2 rotations. The red lines represent the reflection symmetries, where the trianlge is flipped over, while the arrows represent the rotational symmetry of the triangle. -
Order (Group Theory) 1 Order (Group Theory)
Order (group theory) 1 Order (group theory) In group theory, a branch of mathematics, the term order is used in two closely-related senses: • The order of a group is its cardinality, i.e., the number of its elements. • The order, sometimes period, of an element a of a group is the smallest positive integer m such that am = e (where e denotes the identity element of the group, and am denotes the product of m copies of a). If no such m exists, a is said to have infinite order. All elements of finite groups have finite order. The order of a group G is denoted by ord(G) or |G| and the order of an element a by ord(a) or |a|. Example Example. The symmetric group S has the following multiplication table. 3 • e s t u v w e e s t u v w s s e v w t u t t u e s w v u u t w v e s v v w s e u t w w v u t s e This group has six elements, so ord(S ) = 6. By definition, the order of the identity, e, is 1. Each of s, t, and w 3 squares to e, so these group elements have order 2. Completing the enumeration, both u and v have order 3, for u2 = v and u3 = vu = e, and v2 = u and v3 = uv = e. Order and structure The order of a group and that of an element tend to speak about the structure of the group. -
Generic Polynomials Are Descent-Generic
Generic Polynomials are Descent-Generic Gregor Kemper IWR, Universit¨atHeidelberg, Im Neuenheimer Feld 368 69 120 Heidelberg, Germany email [email protected] January 8, 2001 Abstract Let g(X) K(t1; : : : ; tm)[X] be a generic polynomial for a group G in the sense that every Galois extension2 N=L of infinite fields with group G and K L is given by a specialization of g(X). We prove that then also every Galois extension whose≤ group is a subgroup of G is given in this way. Let K be a field and G a finite group. Let us call a monic, separable polynomial g(t1; : : : ; tm;X) 2 K(t1; : : : ; tm)[X] generic for G over K if the following two properties hold. (1) The Galois group of g (as a polynomial in X over K(t1; : : : ; tm)) is G. (2) If L is an infinite field containing K and N=L is a Galois field extension with group G, then there exist λ1; : : : ; λm L such that N is the splitting field of g(λ1; : : : ; λm;X) over L. 2 We call g descent-generic if it satisfies (1) and the stronger property (2') If L is an infinite field containing K and N=L is a Galois field extension with group H G, ≤ then there exist λ1; : : : ; λm L such that N is the splitting field of g(λ1; : : : ; λm;X) over L. 2 DeMeyer [2] proved that the existence of an irreducible descent-generic polynomial for a group G over an infinite field K is equivalent to the existence of a generic extension S=R for G over K in the sense of Saltman [6]. -
CYCLIC RESULTANTS 1. Introduction the M-Th Cyclic Resultant of A
CYCLIC RESULTANTS CHRISTOPHER J. HILLAR Abstract. We characterize polynomials having the same set of nonzero cyclic resultants. Generically, for a polynomial f of degree d, there are exactly 2d−1 distinct degree d polynomials with the same set of cyclic resultants as f. How- ever, in the generic monic case, degree d polynomials are uniquely determined by their cyclic resultants. Moreover, two reciprocal (\palindromic") polyno- mials giving rise to the same set of nonzero cyclic resultants are equal. In the process, we also prove a unique factorization result in semigroup algebras involving products of binomials. Finally, we discuss how our results yield algo- rithms for explicit reconstruction of polynomials from their cyclic resultants. 1. Introduction The m-th cyclic resultant of a univariate polynomial f 2 C[x] is m rm = Res(f; x − 1): We are primarily interested here in the fibers of the map r : C[x] ! CN given by 1 f 7! (rm)m=0. In particular, what are the conditions for two polynomials to give rise to the same set of cyclic resultants? For technical reasons, we will only consider polynomials f that do not have a root of unity as a zero. With this restriction, a polynomial will map to a set of all nonzero cyclic resultants. Our main result gives a complete answer to this question. Theorem 1.1. Let f and g be polynomials in C[x]. Then, f and g generate the same sequence of nonzero cyclic resultants if and only if there exist u; v 2 C[x] with u(0) 6= 0 and nonnegative integers l1; l2 such that deg(u) ≡ l2 − l1 (mod 2), and f(x) = (−1)l2−l1 xl1 v(x)u(x−1)xdeg(u) g(x) = xl2 v(x)u(x): Remark 1.2. -
Finite Abelian P-Primary Groups
CLASSIFICATION OF FINITE ABELIAN GROUPS 1. The main theorem Theorem 1.1. Let A be a finite abelian group. There is a sequence of prime numbers p p p 1 2 ··· n (not necessarily all distinct) and a sequence of positive integers a1,a2,...,an such that A is isomorphic to the direct product ⇠ a1 a2 an A Zp Zp Zpn . ! 1 ⇥ 2 ⇥···⇥ In particular n A = pai . | | i iY=n Example 1.2. We can classify abelian groups of order 144 = 24 32. Here are the possibilities, with the partitions of the powers of 2 and 3 on⇥ the right: Z2 Z2 Z2 Z2 Z3 Z3;(4, 2) = (1 + 1 + 1 + 1, 1 + 1) ⇥ ⇥ ⇥ ⇥ ⇥ Z2 Z2 Z4 Z3 Z3;(4, 2) = (1 + 1 + 2, 1 + 1) ⇥ ⇥ ⇥ ⇥ Z4 Z4 Z3 Z3;(4, 2) = (2 + 2, 1 + 1) ⇥ ⇥ ⇥ Z2 Z8 Z3 Z3;(4, 2) = (1 + 3, 1 + 1) ⇥ ⇥ ⇥ Z16 Z3 Z3;(4, 2) = (4, 1 + 1) ⇥ ⇥ Z2 Z2 Z2 Z2 Z9;(4, 2) = (1 + 1 + 1 + 1, 2) ⇥ ⇥ ⇥ ⇥ Z2 Z2 Z4 Z9;(4, 2) = (1 + 1 + 2, 2) ⇥ ⇥ ⇥ Z4 Z4 Z9;(4, 2) = (2 + 2, 2) ⇥ ⇥ Z2 Z8 Z9;(4, 2) = (1 + 3, 2) ⇥ ⇥ Z16 Z9 cyclic, isomorphic to Z144;(4, 2) = (4, 2). ⇥ There are 10 non-isomorphic abelian groups of order 144. Theorem 1.1 can be broken down into two theorems. 1 2CLASSIFICATIONOFFINITEABELIANGROUPS Theorem 1.3. Let A be a finite abelian group. Let q1,...,qr be the distinct primes dividing A ,andsay | | A = qbj . | | j Yj Then there are subgroups A A, j =1,...,r,with A = qbj ,andan j ✓ | j| j isomorphism A ⇠ A A A . -
Constructive Galois Theory with Linear Algebraic Groups
Constructive Galois Theory with Linear Algebraic Groups Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales LSU REU, Summer 2015 April 14, 2018 Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups If K ⊆ L is a field extension obtained by adjoining all roots of a family of polynomials*, then L=K is Galois, and Gal(L=K) := Aut(L=K) Idea : Families of polynomials ! L=K ! Gal(L=K) Primer on Galois Theory Let K be a field (e.g., Q; Fq; Fq(t)..). Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Idea : Families of polynomials ! L=K ! Gal(L=K) Primer on Galois Theory Let K be a field (e.g., Q; Fq; Fq(t)..). If K ⊆ L is a field extension obtained by adjoining all roots of a family of polynomials*, then L=K is Galois, and Gal(L=K) := Aut(L=K) Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Primer on Galois Theory Let K be a field (e.g., Q; Fq; Fq(t)..). If K ⊆ L is a field extension obtained by adjoining all roots of a family of polynomials*, then L=K is Galois, and Gal(L=K) := Aut(L=K) Idea : Families of polynomials ! L=K ! Gal(L=K) Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Given a finite group G Does there exist field extensions L=K such that Gal(L=K) =∼ G? Inverse Galois Problem Given Galois field extensions L=K Compute Gal(L=K) Eric Chen, J.T. -
MATH 433 Applied Algebra Lecture 19: Subgroups (Continued). Error-Detecting and Error-Correcting Codes. Subgroups Definition
MATH 433 Applied Algebra Lecture 19: Subgroups (continued). Error-detecting and error-correcting codes. Subgroups Definition. A group H is a called a subgroup of a group G if H is a subset of G and the group operation on H is obtained by restricting the group operation on G. Let S be a nonempty subset of a group G. The group generated by S, denoted hSi, is the smallest subgroup of G that contains the set S. The elements of the set S are called generators of the group hSi. Theorem (i) The group hSi is the intersection of all subgroups of G that contain the set S. (ii) The group hSi consists of all elements of the form g1g2 . gk , where each gi is either a generator s ∈ S or the inverse s−1 of a generator. A cyclic group is a subgroup generated by a single element: hgi = {g n : n ∈ Z}. Lagrange’s theorem The number of elements in a group G is called the order of G and denoted o(G). Given a subgroup H of G, the number of cosets of H in G is called the index of H in G and denoted [G : H]. Theorem (Lagrange) If H is a subgroup of a finite group G, then o(G) = [G : H] · o(H). In particular, the order of H divides the order of G. Corollary (i) If G is a finite group, then the order of any element g ∈ G divides the order of G. (ii) If G is a finite group, then g o(G) = 1 for all g ∈ G. -
Chapter 1 GENERAL STRUCTURE and PROPERTIES
Chapter 1 GENERAL STRUCTURE AND PROPERTIES 1.1 Introduction In this Chapter we would like to introduce the main de¯nitions and describe the main properties of groups, providing examples to illustrate them. The detailed discussion of representations is however demanded to later Chapters, and so is the treatment of Lie groups based on their relation with Lie algebras. We would also like to introduce several explicit groups, or classes of groups, which are often encountered in Physics (and not only). On the one hand, these \applications" should motivate the more abstract study of the general properties of groups; on the other hand, the knowledge of the more important and common explicit instances of groups is essential for developing an e®ective understanding of the subject beyond the purely formal level. 1.2 Some basic de¯nitions In this Section we give some essential de¯nitions, illustrating them with simple examples. 1.2.1 De¯nition of a group A group G is a set equipped with a binary operation , the group product, such that1 ¢ (i) the group product is associative, namely a; b; c G ; a (b c) = (a b) c ; (1.2.1) 8 2 ¢ ¢ ¢ ¢ (ii) there is in G an identity element e: e G such that a e = e a = a a G ; (1.2.2) 9 2 ¢ ¢ 8 2 (iii) each element a admits an inverse, which is usually denoted as a¡1: a G a¡1 G such that a a¡1 = a¡1 a = e : (1.2.3) 8 2 9 2 ¢ ¢ 1 Notice that the axioms (ii) and (iii) above are in fact redundant.