DOCSLIB.ORG

Parity Alternating Permutations Starting with an Odd Integer

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Parity alternating permutations starting with an odd integer Frether Getachew Kebede1a, Fanja Rakotondrajaob aDepartment of Mathematics, College of Natural and Computational Sciences, Addis Ababa University, P.O.Box 1176, Addis Ababa, Ethiopia; e-mail: [email protected] bD´epartement de Math´ematiques et Informatique, BP 907 Universit´ed’Antananarivo, 101 Antananarivo, Madagascar; e-mail: [email protected] Abstract A Parity Alternating Permutation of the set [n] = 1, 2,...,n is a permutation with { } even and odd entries alternatively. We deal with parity alternating permutations having an odd entry in the first position, PAPs. We study the numbers that count the PAPs with even as well as odd parity. We also study a subclass of PAPs being derangements as well, Parity Alternating Derangements (PADs). Moreover, by considering the parity of these PADs we look into their statistical property of excedance. Keywords: parity, parity alternating permutation, parity alternating derangement, excedance 2020 MSC: 05A05, 05A15, 05A19 arXiv:2101.09125v1 [math.CO] 22 Jan 2021 1. Introduction and preliminaries A permutation π is a bijection from the set [n]= 1, 2,...,n to itself and we will write it { } in standard representation as π = π(1) π(2) π(n), or as the product of disjoint cycles. ··· The parity of a permutation π is defined as the parity of the number of transpositions (cycles of length two) in any representation of π as a product of transpositions. One 1Corresponding author. n c way of determining the parity of π is by obtaining the sign of ( 1) − , where c is the − number of cycles in the cycle representation of π. That is, if the sign of π is -1, then π is called an odd permutation, and an even permutation otherwise. For example, the permutation 42178635= (1 4 7 3)(2)(5 8)(6), of length 8, is even since it has sign 1. All basic definitions and properties not explained here can be found in for example [8] and [4]. According to [9], a Parity Alternating Permutation over the set [n] is a permutation, in standard form, with even and odd entries alternatively (in this general sense). The set of all parity alternating permutations is a subgroup of the symmetric group S , the Pn n group of all permutations over [n]. The order of the set set has been studied lately Pn in relations to other number sequences such as Eulerian numbers (see [10, 9]). However, in this paper we will deal only with the parity alternating permutations which in addition have an odd entry in the first position; and we call them PAPs. It can be shown that the set Pn containing all PAPs over [n] is a subgroup of the symmetric group S and also of the group . We consider this kind of permutations because, n Pn for odd n there are no parity alternating permutations over [n] beginning with an even integer. Avi Peretz determined the number sequence that count the number of PAPs (see A010551). Unfortunately, we could not find any details of his work. In A010551, we can also find the exponential generating function of these numbers due to Paul D. Hanna. Since there is no published proof of this formula we prove it here, as Theorem 2.2. Moreover, the numbers that count the PAPs with even parity and with odd parity (which were not studied before) are determined. By pn we denote the cardinality of the set Pn of all PAPs over [n]. Let φn denote a map n n from Pn to S S that relates a PAP σ to a pair of permutations (σ1, σ2) in the set ⌈ 2 ⌉ × ⌊ 2 ⌋ n n σ(2i 1)+1 σ(2i) S S , in such a way that σ1(i)= − and σ2(i)= . It is easy to see that ⌈ 2 ⌉ × ⌊ 2 ⌋ 2 2 this map is a bijection. For example, the PAPs 5 2 1 4 3 6 7 and 7 4 5 6 3 2 1 over [7] are mapped to the pairs (3124, 123) and (4321, 2 3 1), respectively. If we consider a PAP 2 σ in cycle representations, then each cycle consists of integers of the same parity. Thus, we immediately get cycle representation of σ1 and σ2. For instance, the cycle form of the two PAPs above are (153)(7)(2)(4)(6) and (17)(35)(246) which correspond to the pairs ((132)(4), (1)(2)(3)) and ((14)(23), (1 2 3)), respectively. (Unless stated otherwise we will always use (disjoint) cycle representation of permutations.) Another way of looking at the mapping φn is that σ1 and σ2 correspond to the parts that contain the odd and even integers in σ, respectively. Therefore, studying PAPs is similar to studying the two permutations that correspond to the even and the odd integers in the PAP separately and then combining the properties. In Table 1, we give a short summary of properties that permutations and PAPs satisfy (for detailed discussions, see Section 2). Permutations PAPs Seq 1, 1, 2, 6, 24, 120,.... (A000142) 1, 1, 1, 2, 4, 12,.... (A010551) 2√4 x2+2 cos−1(1 x2/2) EGF 1 − − 1 x (2 x)√4 x2 − − − Even (seq) 1, 1, 1, 3, 12, 60,.... (A001710) 1, 1, 1, 1, 2, 6, 18, 72,... Odd (seq) 1, 1, 1, 3, 12, 60,.... (A001710) 0, 0, 0, 1, 2, 6, 18, 72,... x2 √4 x2+cos−1 1 2 x2 − − 2 x2 x 1 Even (EGF) − + + + 2 2x (2 x)√4 x2 4 2 2 − − − 2 2 −1 x 2 √4 x +cos 1 2 Odd (EGF) x − − 2 x x 1 2 2x (2 x)√4 x2 4 2 2 − − − − − − Table 1: A comparison table of permutations and PAPs (EGF mean exponential generating function). One interesting subset of S is the set D of derangements. For d = D , we have a n n n | n| well known relation dn =(n 1)[dn 1 + dn 2], d0 = 1 and d1 =0 (1) − − − for n 2. A proof of this relation may be found in any textbook on combinatorics, but ≥ we will have later use of the following bijection due to Mantaci and Rakotondrajao ([6]). They define ψn to be the bijection between Dn and [n 1] (Dn 1 Dn 2) as follows: − × − ∪ − 3 (1) let Dn denote the set of derangements over [n] having the integer n in a cycle of length (2) greater than 2, and Dn be the set of derangements over [n] having n in a transposition. These two sets are disjoint and their union is D . Then for δ D define ψ (δ)=(i, δ′), n ∈ n n 1 where i = δ− (n) and δ′ is the derangement obtained from (1) δ Dn by removing n or • ∈ (2) δ Dn by removing the transposition (i n) and then decreasing all integers • ∈ greater than i by 1. For instance, the pairs (2, (152)(34)) and (2, (1 2)(3 4)) correspond to the derangements (1526)(34) and (13)(45)(26), respectively, for n = 6. We denote the restricted bijec- (1) (2) tions ψn (1) and ψn (2) by ψn and ψn , respectively. |Dn |Dn Another important, and more difficult to prove, recurrence relation that the numbers dn satisfy is n dn = ndn 1 +( 1) , d0 =0 (2) − − for n 1. We will later make a use of the bijection τn : ([n] Dn 1) Fn Dn En ≥ × − \ −→ \ given by the second author ([7]) proving the recurrence. Where En is the set containing the derangement ∆ = (12)(34) (n 1 n) for even n, and is empty for odd n. F n ··· − n is the set containing the pair (n, ∆n 1) when n is odd, and is empty when n is even. − Thus, the inverse ζn of τn relates an element of [n 1] Dn 1 with every derangement − × − over [n] that has the integer n in a cycle of length greater than 2, and an element of n Dn 1 Fn with every derangement over [n] in which n lies in a transposition. { } × − \ Classifying derangements by their parity, we denote the number of even and odd de- e o e o rangements over [n] by dn and dn, respectively. Clearly dn = dn + dn. Moreover, the e o numbers dn and dn satisfy the relations e o o o e e dn =(n 1)[dn 1 + dn 2] and dn =(n 1)[dn 1 + dn 2], (3) − − − − − − for n 2 with initial conditions de = 1, de = 0, do = 0, and do = 0 ([6], Proposition ≥ 0 1 0 1 4.1). 4 We will put a major interest on Parity Alternating Derangements (PADs) which are the derangements which also are parity alternating permutations starting with odd integers. Let d denote cardinality of the set of PADs D = D P . The restricted bijection n n n ∩ n n n Φn = φn Dn : Dn D D will let us consider the odd parts and the even parts of | −→ ⌈ 2 ⌉ × ⌊ 2 ⌋ any given PAD regarded as ordinary derangements with smaller length than the length of the PAD. The mapping Φn plays the central role in our investigations. In Table 2, we display the connection of ordinary derangements and PADs (for detailed discussions, see Section 3). Finding explicit expressions for some of the generating functions are still open questions. On the other hand the EGF for the PADs for example is the solution to an eighth order differential equation with polynomial coefficients, and also is expressible in terms of Hadamard products of some known generating functions. Derangements PADs Seq 1, 0, 1, 2, 9, 44,... (A000166) 1, 0, 0, 0, 1, 2, 4, 18, 81, 396,..
Recommended publications
  • Abstract Algebra

    Abstract Algebra

    Abstract Algebra Paul Melvin Bryn Mawr College Fall 2011 lecture notes loosely based on Dummit and Foote's text Abstract Algebra (3rd ed) Prerequisite: Linear Algebra (203) 1 Introduction Pure Mathematics Algebra Analysis Foundations (set theory/logic) G eometry & Topology What is Algebra? • Number systems N = f1; 2; 3;::: g \natural numbers" Z = f:::; −1; 0; 1; 2;::: g \integers" Q = ffractionsg \rational numbers" R = fdecimalsg = pts on the line \real numbers" p C = fa + bi j a; b 2 R; i = −1g = pts in the plane \complex nos" k polar form re iθ, where a = r cos θ; b = r sin θ a + bi b r θ a p Note N ⊂ Z ⊂ Q ⊂ R ⊂ C (all proper inclusions, e.g. 2 62 Q; exercise) There are many other important number systems inside C. 2 • Structure \binary operations" + and · associative, commutative, and distributive properties \identity elements" 0 and 1 for + and · resp. 2 solve equations, e.g. 1 ax + bx + c = 0 has two (complex) solutions i p −b ± b2 − 4ac x = 2a 2 2 2 2 x + y = z has infinitely many solutions, even in N (thei \Pythagorian triples": (3,4,5), (5,12,13), . ). n n n 3 x + y = z has no solutions x; y; z 2 N for any fixed n ≥ 3 (Fermat'si Last Theorem, proved in 1995 by Andrew Wiles; we'll give a proof for n = 3 at end of semester). • Abstract systems groups, rings, fields, vector spaces, modules, . A group is a set G with an associative binary operation ∗ which has an identity element e (x ∗ e = x = e ∗ x for all x 2 G) and inverses for each of its elements (8 x 2 G; 9 y 2 G such that x ∗ y = y ∗ x = e).
  • Arxiv:2009.00554V2 [Math.CO] 10 Sep 2020 Graph On√ More Than Half of the Vertices of the D-Dimensional Hypercube Qd Has Maximum Degree at Least D

    Arxiv:2009.00554V2 [Math.CO] 10 Sep 2020 Graph On√ More Than Half of the Vertices of the D-Dimensional Hypercube Qd Has Maximum Degree at Least D

    ON SENSITIVITY IN BIPARTITE CAYLEY GRAPHS IGNACIO GARCÍA-MARCO Facultad de Ciencias, Universidad de La Laguna, La Laguna, Spain KOLJA KNAUER∗ Aix Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Spain ABSTRACT. Huang proved that every set of more than halfp the vertices of the d-dimensional hyper- cube Qd induces a subgraph of maximum degree at least d, which is tight by a result of Chung, Füredi, Graham, and Seymour. Huang asked whether similar results can be obtained for other highly symmetric graphs. First, we present three infinite families of Cayley graphs of unbounded degree that contain in- duced subgraphs of maximum degree 1 on more than half the vertices. In particular, this refutes a conjecture of Potechin and Tsang, for which first counterexamples were shown recently by Lehner and Verret. The first family consists of dihedrants and contains a sporadic counterexample encoun- tered earlier by Lehner and Verret. The second family are star graphs, these are edge-transitive Cayley graphs of the symmetric group. All members of the third family are d-regular containing an d induced matching on a 2d−1 -fraction of the vertices. This is largest possible and answers a question of Lehner and Verret. Second, we consider Huang’s lower bound for graphs with subcubes and show that the corre- sponding lower bound is tight for products of Coxeter groups of type An, I2(2k + 1), and most exceptional cases. We believe that Coxeter groups are a suitable generalization of the hypercube with respect to Huang’s question.
  • On the Parity of the Number of Multiplicative Partitions and Related Problems

    On the Parity of the Number of Multiplicative Partitions and Related Problems

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 11, November 2012, Pages 3793–3803 S 0002-9939(2012)11254-7 Article electronically published on March 15, 2012 ON THE PARITY OF THE NUMBER OF MULTIPLICATIVE PARTITIONS AND RELATED PROBLEMS PAUL POLLACK (Communicated by Ken Ono) Abstract. Let f(N) be the number of unordered factorizations of N,where a factorization is a way of writing N as a product of integers all larger than 1. For example, the factorizations of 30 are 2 · 3 · 5, 5 · 6, 3 · 10, 2 · 15, 30, so that f(30) = 5. The function f(N), as a multiplicative analogue of the (additive) partition function p(N), was first proposed by MacMahon, and its study was pursued by Oppenheim, Szekeres and Tur´an, and others. Recently, Zaharescu and Zaki showed that f(N) is even a positive propor- tion of the time and odd a positive proportion of the time. Here we show that for any arithmetic progression a mod m,thesetofN for which f(N) ≡ a(mod m) possesses an asymptotic density. Moreover, the density is positive as long as there is at least one such N. For the case investigated by Zaharescu and Zaki, we show that f is odd more than 50 percent of the time (in fact, about 57 percent). 1. Introduction Let f(N) be the number of unordered factorizations of N,whereafactorization of N is a way of writing N as a product of integers larger than 1. For example, f(12) = 4, corresponding to 2 · 6, 2 · 2 · 3, 3 · 4, 12.
  • The ℓ-Parity Conjecture Over the Constant Quadratic Extension

    The ℓ-Parity Conjecture Over the Constant Quadratic Extension

    THE `-PARITY CONJECTURE OVER THE CONSTANT QUADRATIC EXTENSION KĘSTUTIS ČESNAVIČIUS Abstract. For a prime ` and an abelian variety A over a global field K, the `-parity conjecture predicts that, in accordance with the ideas of Birch and Swinnerton-Dyer, the Z`-corank of the `8-Selmer group and the analytic rank agree modulo 2. Assuming that char K ¡ 0, we prove that the `-parity conjecture holds for the base change of A to the constant quadratic extension if ` is odd, coprime to char K, and does not divide the degree of every polarization of A. The techniques involved in the proof include the étale cohomological interpretation of Selmer groups, the Grothendieck–Ogg–Shafarevich formula, and the study of the behavior of local root numbers in unramified extensions. 1. Introduction 1.1. The `-parity conjecture. The Birch and Swinnerton-Dyer conjecture (BSD) predicts that the completed L-function LpA; sq of an abelian variety A over a global field K extends meromorphically to the whole complex plane, vanishes to the order rk A at s “ 1, and satisfies the functional equation LpA; 2 ´ sq “ wpAqLpA; sq; (1.1.1) where rk A and wpAq are the Mordell–Weil rank and the global root number of A. The vanishing assertion combines with (1.1.1) to give “BSD modulo 2,” namely, the parity conjecture: ? p´1qrk A “ wpAq: Selmer groups tend to be easier to study than Mordell–Weil groups, so, fixing a prime `, one sets 8 rk` A :“ dimQ` HompSel` A; Q`{Z`q bZ` Q`; where Sel 8 A : lim Sel n A ` “ ÝÑ ` is the `8-Selmer group, notes that the conjectured finiteness of the Shafarevich–Tate group XpAq implies rk` A “ rk A, and instead of the parity conjecture considers the `-parity conjecture: ? p´1qrk` A “ wpAq: (1.1.2) 1.2.
  • Notes on Finite Group Theory

    Notes on Finite Group Theory

    Notes on finite group theory Peter J. Cameron October 2013 2 Preface Group theory is a central part of modern mathematics. Its origins lie in geome- try (where groups describe in a very detailed way the symmetries of geometric objects) and in the theory of polynomial equations (developed by Galois, who showed how to associate a finite group with any polynomial equation in such a way that the structure of the group encodes information about the process of solv- ing the equation). These notes are based on a Masters course I gave at Queen Mary, University of London. Of the two lecturers who preceded me, one had concentrated on finite soluble groups, the other on finite simple groups; I have tried to steer a middle course, while keeping finite groups as the focus. The notes do not in any sense form a textbook, even on finite group theory. Finite group theory has been enormously changed in the last few decades by the immense Classification of Finite Simple Groups. The most important structure theorem for finite groups is the Jordan–Holder¨ Theorem, which shows that any finite group is built up from finite simple groups. If the finite simple groups are the building blocks of finite group theory, then extension theory is the mortar that holds them together, so I have covered both of these topics in some detail: examples of simple groups are given (alternating groups and projective special linear groups), and extension theory (via factor sets) is developed for extensions of abelian groups. In a Masters course, it is not possible to assume that all the students have reached any given level of proficiency at group theory.
  • Maxmalgebra M the Linear Algebra of Combinatorics?

    Maxmalgebra M the Linear Algebra of Combinatorics?

    Max-algebra - the linear algebra of combinatorics? Peter Butkoviÿc School of Mathematics and Statistics The University of Birmingham Edgbaston Birmingham B15 2TT United Kingdom Submitted by ABSTRACT Let a b =max(a, b),a b = a + b for a, b R := R .Bymax- algebra we⊕ understand the analogue⊗ of linear algebra∈ developed∪ {−∞ for} the pair of operations ( , ) extended to matrices and vectors. Max-algebra, which has been studied⊕ for⊗ more than 40 years, is an attractive way of describing a class of nonlinear problems appearing for instance in machine-scheduling, information technology and discrete-event dynamic systems. This paper focuses on presenting a number of links between basic max-algebraic problems like systems of linear equations, eigenvalue-eigenvector problem, linear independence, regularity and characteristic polynomial on one hand and combinatorial or combinatorial opti- misation problems on the other hand. This indicates that max-algebra may be regarded as a linear-algebraic encoding of a class of combinatorial problems. The paper is intended for wider readership including researchers not familiar with max-algebra. 1. Introduction Let a b =max(a, b) ⊕ and a b = a + b ⊗ for a, b R := R . By max-algebra we understand in this paper the analogue∈ of linear∪ {−∞ algebra} developed for the pair of operations ( , ) extended to matrices and vectors formally in the same way as in linear⊕ ⊗ algebra, that is if A =(aij),B=(bij) and C =(cij ) are matrices with 1 2 elements from R of compatible sizes, we write C = A B if cij = aij bij ⊕ ⊕ for all i, j, C = A B if cij = ⊕ aik bkj =maxk(aik + bkj) for all i, j ⊗ k ⊗ and α A =(α aij) for α R.
  • Some Problems of Erdős on the Sum-Of-Divisors Function

    Some Problems of Erdős on the Sum-Of-Divisors Function

    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, SERIES B Volume 3, Pages 1–26 (April 5, 2016) http://dx.doi.org/10.1090/btran/10 SOME PROBLEMS OF ERDOS˝ ON THE SUM-OF-DIVISORS FUNCTION PAUL POLLACK AND CARL POMERANCE For Richard Guy on his 99th birthday. May his sequence be unbounded. Abstract. Let σ(n) denote the sum of all of the positive divisors of n,and let s(n)=σ(n) − n denote the sum of the proper divisors of n. The functions σ(·)ands(·) were favorite subjects of investigation by the late Paul Erd˝os. Here we revisit three themes from Erd˝os’s work on these functions. First, we improve the upper and lower bounds for the counting function of numbers n with n deficient but s(n) abundant, or vice versa. Second, we describe a heuristic argument suggesting the precise asymptotic density of n not in the range of the function s(·); these are the so-called nonaliquot numbers. Finally, we prove new results on the distribution of friendly k-sets, where a friendly σ(n) k-set is a collection of k distinct integers which share the same value of n . 1. Introduction Let σ(n) be the sum of the natural number divisors of n,sothatσ(n)= d|n d. Let s(n) be the sum of only the proper divisors of n,sothats(n)=σ(n) − n. Interest in these functions dates back to the ancient Greeks, who classified numbers as deficient, perfect,orabundant according to whether s(n) <n, s(n)=n,or s(n) >n, respectively.
  • Gender Equality: Glossary of Terms and Concepts

    Gender Equality: Glossary of Terms and Concepts

    GENDER EQUALITY: GLOSSARY OF TERMS AND CONCEPTS GENDER EQUALITY Glossary of Terms and Concepts UNICEF Regional Office for South Asia November 2017 Rui Nomoto GENDER EQUALITY: GLOSSARY OF TERMS AND CONCEPTS GLOSSARY freedoms in the political, economic, social, a cultural, civil or any other field” [United Nations, 1979. ‘Convention on the Elimination of all forms of Discrimination Against Women,’ Article 1]. AA-HA! Accelerated Action for the Health of Adolescents Discrimination can stem from both law (de jure) or A global partnership, led by WHO and of which from practice (de facto). The CEDAW Convention UNICEF is a partner, that offers guidance in the recognizes and addresses both forms of country context on adolescent health and discrimination, whether contained in laws, development and puts a spotlight on adolescent policies, procedures or practice. health in regional and global health agendas. • de jure discrimination Adolescence e.g., in some countries, a woman is not The second decade of life, from the ages of 10- allowed to leave the country or hold a job 19. Young adolescence is the age of 10-14 and without the consent of her husband. late adolescence age 15-19. This period between childhood and adulthood is a pivotal opportunity to • de facto discrimination consolidate any loss/gain made in early e.g., a man and woman may hold the childhood. All too often adolescents - especially same job position and perform the same girls - are endangered by violence, limited by a duties, but their benefits may differ. lack of quality education and unable to access critical health services.i UNICEF focuses on helping adolescents navigate risks and vulnerabilities and take advantage of e opportunities.
  • Algebra Course Notes

    Algebra Course Notes

    Algebra course notes Margaret Bilu December 20, 2019 1 Contents 1 Quantifiers, sets, maps, equivalence relations 3 1.1 Sets, subsets . 3 1.2 Unions, intersections, products . 4 1.3 Maps . 5 1.4 Equivalence relations . 5 2 Integers 7 2.1 Addition and multiplication on the set of integers Z ............. 7 2.2 Divisibility . 8 2.3 Euclidean division . 9 2.4 GCD and Euclid’s algorithm . 9 2.5 Unique factorization of integers . 10 2.6 Congruence classes . 11 2.7 Units in Z=nZ .................................. 13 2.8 Back to congruences . 14 2.9 Conclusion of the chapter . 15 3 Groups 15 3.1 Laws of composition . 15 3.2 Groups . 17 3.3 Subgroups . 19 3.4 Products of groups . 20 3.5 Cyclic groups . 20 3.6 Group homomorphisms . 22 3.7 Isomorphisms . 24 3.8 Classification of groups of small order . 26 3.9 Conclusion of the chapter . 27 4 Permutation groups 28 4.1 Definition . 29 4.2 Cycles . 30 4.3 Parity of a permutation . 32 4.4 Generators of Sn and An ............................ 34 4.5 Dihedral groups . 35 4.6 Conclusion of the chapter . 36 5 Cosets and Lagrange’s theorem 37 5.1 Left and right cosets . 37 5.2 Index of a subgroup . 38 5.3 Lagrange’s theorem . 40 5.4 Some arithmetic applications of Lagrange’s theorem . 42 2 5.5 Cosets and homomorphisms . 43 5.6 Conclusion of the chapter . 45 6 Normal subgroups and quotients of groups 46 6.1 Normal subgroups . 46 6.2 Quotient groups .
  • The Alternating Group

    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.
  • Chapter IV. Determinants

    Chapter IV. Determinants

    Notes ⃝c F.P. Greenleaf 2014 LAI-f14-det.tex version 8/11/2014 Chapter IV. Determinants. IV.1 The Permutation Group Sn. The permutation group Sn consists of all bijections σ :[1,n] [1,n]where[1,n]= 1,...,n ,withcompositionofoperators → { } σ σ (k)=σ (σ (k)) for 1 k n 1 ◦ 2 1 2 ≤ ≤ as the group operation. The identity element e is the identity map id[1,n] such that e(k)=k,forallk [1,n]. We recall that a group is any set G equipped with a binary operation ( )satisfyingthefollowingaxioms:∈ ∗ 1. Associativity: x (y z)=(x y) z; ∗ ∗ ∗ ∗ 2. Identity element: There is an e G such that e x = x = x e,forallx G; ∈ ∗ ∗ ∈ 3. Inverses: Every x G has a “two-sided inverse,” an element x−1 G such that x−1 x = x x−1 =∈e. ∈ ∗ ∗ We do not assume that the system (G, )iscommutative,withx y = y x;agroupwith this extra property is a commutative group∗ ,alsoreferredtoasan∗ abelian∗ group. Here are some examples of familiar groups. 1. The integers (Z, +) become a commutative group when equipped with (+) as the group operation; multiplication ( ) does not make Z agroup.(Why?) · 2. Any vector space equipped with its (+) operation is a commutatve group, for instance (Kn, +); 3. The set (C×, )=C 0 of nonzero complex numbers equipped with complex multiplication· ( )isacommutativegroup;soisthesubset∼{} S1 = z C : z =1 (unit circle in the· complex plane) because z , w =1 zw ={ z∈ w =1and| | } 1/z =1/ z =1. | | | | ⇒| | | |·| | | | | | 4. General Linear Group. The set GL(n, K)= A M(n, K):det(A) =0 of invertible n n matrices is a group when equipped{ with∈ matrix multiply̸ as} the group operation.× It is noncommutative when n 2.
  • Introduction to Groups, Rings and Fields

    Introduction to Groups, Rings and Fields

    Introduction to Groups, Rings and Fields HT and TT 2011 H. A. Priestley 0. Familiar algebraic systems: review and a look ahead. GRF is an ALGEBRA course, and specifically a course about algebraic structures. This introduc- tory section revisits ideas met in the early part of Analysis I and in Linear Algebra I, to set the scene and provide motivation. 0.1 Familiar number systems Consider the traditional number systems N = 0, 1, 2,... the natural numbers { } Z = m n m, n N the integers { − | ∈ } Q = m/n m, n Z, n = 0 the rational numbers { | ∈ } R the real numbers C the complex numbers for which we have N Z Q R C. ⊂ ⊂ ⊂ ⊂ These come equipped with the familiar arithmetic operations of sum and product. The real numbers: Analysis I built on the real numbers. Right at the start of that course you were given a set of assumptions about R, falling under three headings: (1) Algebraic properties (laws of arithmetic), (2) order properties, (3) Completeness Axiom; summarised as saying the real numbers form a complete ordered field. (1) The algebraic properties of R You were told in Analysis I: Addition: for each pair of real numbers a and b there exists a unique real number a + b such that + is a commutative and associative operation; • there exists in R a zero, 0, for addition: a +0=0+ a = a for all a R; • ∈ for each a R there exists an additive inverse a R such that a+( a)=( a)+a = 0. • ∈ − ∈ − − Multiplication: for each pair of real numbers a and b there exists a unique real number a b such that · is a commutative and associative operation; • · there exists in R an identity, 1, for multiplication: a 1 = 1 a = a for all a R; • · · ∈ for each a R with a = 0 there exists an additive inverse a−1 R such that a a−1 = • a−1 a = 1.∈ ∈ · · Addition and multiplication together: forall a,b,c R, we have the distributive law a (b + c)= a b + a c.