FINITE GROUPS WITH THREE RELATIVE COMMUTATIVITY DEGREES. Communicated by Ali Reza Ashrafi. 1. Introduction
|
|
- Ellen Johnston
- 6 years ago
- Views:
Transcription
1 Bulleti of the Iraia Mathematical Society Vol. 39 No ), pp FINITE GROUPS WITH THREE RELATIVE COMMUTATIVITY DEGREES R. BARZGAR, A. ERFANIAN AND M. FARROKHI D. G. Commuicated by Ali Reza Ashrafi Abstract. For a fiite group G ad a subgroup H of G, the relative commutativity degree of H i G, deoted by dh, G), is the probability that a elemet of H commutes with a elemet of G. Let DG) = {dh, G) : H G} be the set of all relative commutativity degrees of subgroups of G. It is show that a fiite group G admits three relative commutativity degrees if ad oly if G/ZG) is a o-cyclic group of order pq, where p ad q are primes. Moreover, we determie all the relative commutativity degrees of some kow groups.. Itroductio If G is a fiite group, the the commutativity degree of G, deoted by dg), is the probability that two radomly chose elemets of G commute. The commutativity degree first studied by Gustafso [4] ad it was show that dg) 5 8 for every o-abelia fiite group G ad equality holds precisely whe G/ZG) = Z 2 Z 2. Let H be a subgroup of G. Erfaia et al. i [] geeralized the commutativity degree of G by cosiderig the relative commutativity MSC200): Primary: 20P05; Secodary: 20E45. Keywords: Commutativity degree, relative commutativity degree, isocliism, relative isocliism. Received: 8 July 20, Accepted: 2 April 202. Correspodig author c 203 Iraia Mathematical Society. 27
2 272 Barzegar, Erfaia ad Farrokhi degree of H i G, deoted by dh, G), as the probability that a elemet of H commutes with a elemet of G that is {h, g) H G : [h, g] = } dh, G) =. G Also they have give several lower ad upper bouds for the relative commutativity degree dh, G). We refer the reader to [, 4, 5] for more details. Now let DG) = {dh, G) : H G}. It is obvious that DG) = if ad oly if G is a abelia group. Also, it is easy to see that there is o fiite group G with DG) = 2 see Lemma 2.4). We ited to study the set DG) ad classify all fiite groups G with three commutativity degrees. We will show that if G is a fiite group, the DG) = 3 if ad oly if G/ZG) is a group of order pq for some primes p ad q. Moreover, the umber of relative commutativity degrees will be computed for some classes of fiite groups icludig dihedral groups, geeralized quaterio groups ad quasi-dihedral groups. The motivatio to the research is the Farrokhi s classificatio of fiite groups with two subgroup ormality degrees i [3]. For further details o this topic, we refer the iterested reader to [6]. 2. Prelimiary results We begi with some basic lemmas. Lemma 2.. Let G be a fiite group ad H K G. The dk, G) dh, G) ad the equality holds if ad oly if K = HC K g) for all g G. Proof. Sice g H g K, we have C Kg) K dk, G) = C K g) G K G g G C Hg) g G for each g G. Hece C H g) Also dk, G) = dh, G) if ad oly if C Kg) K which is equivalet to K = HC K g) for all g G. Note that for ay subgroup H G, we have dh, G) = C H g). G g G = C Hg) = dh, G). for all g G, Lemma 2.2. Let G be a o-abelia fiite group ad x G \ ZG). The d x, G), dg).
3 Relative commutativity degrees 273 Proof. Clearly d x, G) because x is ot cetral. Suppose that d x, G) = dg), the by Lemma 2., G = x C G y) for all y G. I particular G = x C G x) = C G x), which implies that x ZG), a cotradictio. From the above lemma the followig result ca be obtaied immediately. Corollary 2.3. If G is a o-abelia fiite group, the DG) 2. I the sequel we shall discuss fiite groups with three relative commutativity degrees. Lemma 2.4. Let G be a o-abelia fiite group ad suppose that DG) = {, d, dg)}. If H is a subgroup of G such that dh, G) = d, the H is abelia. Proof. Let h H \ ZH), the by Lemma 2.2, d h, G) dg). Thus d h, G) = dh, G) ad by Lemma 2., H = h C H g) for all g G. Now by replacig g by h, we coclude that h ZH), a cotradictio. Lemma 2.5. Let G be a fiite group with DG) = 3. The C G x) is a abelia maximal subgroup of G, for all x G \ ZG). Proof. Let x G \ ZG). If C G x) is a o-abelia group, the by Lemmas 2.4 ad 2., we have G = C G x)c G g) for all g G. I particular, G = C G x)c G x) = C G x) ad hece x ZG), which is a cotradictio. Now let M be a maximal subgroup of G cotaiig C G x). If M is o-abelia, the by Lemmas 2.4 ad 2., G = MC G x) so G = M, a cotradictio. Thus M is abelia ad cosequetly M = C G x), as required. Lemma 2.6. If H is a subgroup of G, the dhzg), G) = dh, G). Proof. The proof is straightforward. Lemma 2.7. Let G be a fiite group with DG) = 3. The the followig assertios are true: i) if x is a p-elemet, the x p ZG); ii) if x, y G\ZG) are p-elemet ad q-elemet, respectively, the p q) G = qp ) C G x) pq ) C G y). I particular, C G x) = C G y) if ad oly if p = q.
4 274 Barzegar, Erfaia ad Farrokhi Proof. i) Let x G \ ZG) be a p-elemet of order p. We proceed by iductio o. Cearly the result holds for = 0,. Suppose that the result holds for. The x p = p ad by hypothesis x p2 ZG). If x p ZG), the by Lemma 2.2, d x, G) = d x p, G). O the other had, d x, G) = p G p 2 G +p p 2 ) C G x p ) +p p ) C G x) ) ad d x p, G) = from which we obtai p G p 2 G + p p 2 ) C G x p ) ), p = [G : C G x)] + p )[C G x p ) : C G x)]. Hece [G : C G x)] = [C G x p ) : C G x)] = that is x ZG), cotradictig the hypothesis. ii) By Lemma 2.2, d x, G) = d y, G). By part i) we have ad d x, G) = p m G pm G + p m p m ) C G x) ) d y, G) = q G q G + q q ) C G y) ), from which the result follows. The rest of the proof is a direct cosequece of the last two equatios. 3. Mai results We are ow i a positio to give the mai theorems. The ilpotet ad o-ilpotet cases are discussed separately. Theorem 3.. Let G be a fiite ilpotet group. The DG) = 3 if ad oly if G/ZG) = Z p Z p. I particular, DG) = {, 2p, p2 +p }. p 2 p 3 Proof. Sice G is ilpotet, so G = P P 2 P, where P i is the Sylow p i -subgroup of G. Clearly DG) = DP )DP 2 ) DP ) DP ) DP 2 ) DP ). Sice DG) = 3, there exists a Sylow p i -subgroup P = P i such that DP ) = DG) ad P j is abelia for each j i. Let x, y P such that xy yx, M = C P x) ad N = C P y). The by Lemma 2.5i),
5 Relative commutativity degrees 275 M ad N are abelia maximal subgroups of P. Clearly P = MN ad M N = ZP ). Hece P ZP ) = MN M N = M M N N M N = p2. Coversely, let H G be a proper o-cetral subgroup of G such that dh, G) dg). By Lemma 2.6, dh, G) = dhzg), G) ad we may assume that ZG) H. The H/ZG) = Z p ad we have dh, G) = G = p ZG) = 2p p 2. h H C G h) = h H h G ) ZG) + p ) ZG) p Similarly, it ca be easily show that dg) = p2 +p p 3 ad cosequetly DG) = {, 2p, p2 +p }. p 2 p 3 Theorem 3.2. Let G be a fiite o-ilpotet group. The DG) = 3 if ad oly if G/ZG) is a o-cyclic group of order pq, where p ad q are distict primes. I particular, DG) = {, p + q pq, p + }, q 2 pq 2 wheever p > q. Proof. Sice G is ot ilpotet, there exist a p-elemet x ad a q-elemet y p q) such that xy yx. Clearly we may assume that q is the smallest prime dividig G/ZG). By Lemma 2.5, M = C G x) ad N = C G y) are differet abelia maximal subgroups of G. Moreover, M N = ZG) ad by Lemma ) p q) G = qp ) M pq ) N. Note that by Lemma 2.7ii), M N so that M ad N are ot cojugate. Let : G G/ZG) be the atural homomorphism. If M ad N are o-ormal subgroups of G, the N G M) = M, N G N) = N ad the
6 276 Barzegar, Erfaia ad Farrokhi cojugates of M ad N all have trivial itersectio. Thus G M g N g g G g G = + [G : M] M ) + [G : N] N ) + G 2 + G 2 > G, which is a cotradictio. Thus M or N is a ormal subgroup of G, which implies that G = MN. If M, N G, the G = M N is abelia ad hece G is ilpotet, a cotradictio. Therefore G is a Frobeius group with M ad N as its Frobeius kerel ad complemet i some order. Moreover, gcd M, N ) =. From the equatio 3.) it follows that M divides pq ) N, which implies that M divides p. Hece M = p ad, usig the equatio 3.) oce more, we obtai N = q. Therefore G = pq, as required. Coversely, suppose that G is a fiite o-ilpotet group such that G/ZG) = pq for some primes. Clearly G/ZG) is o-abelia, p q ad we may assume that p > q. Let H be a proper o-cetral subgroup of G. The by Lemma 2.6, we may assume that ZG) H. Thus H/ZG) = Z p or Z q. I the first case dh, G) = ad i the secod case dh, G) = ZG) + p ) ZG) q ZG) + q ) ZG) p ) = p + q pq ) = p + q pq. O the other had G/ZG) is a Frobeius group with Frobeius kerel ad complemet isomorphic to cyclic groups of orders p ad q, respectively. Thus dg) = G ZG) + p ) ZG) ) q + pq p) ZG) p = p + q 2 pq 2. Therefore DG) = {, p + q pq, p + q 2 pq 2 } ad the proof is complete.
7 Relative commutativity degrees Examples This subsectio is devoted to the determiatio of the set of all relative commutativity degrees of some classes of isocliic fiite groups. First, we compute the set of all relative commutativity degrees of dihedral groups, the we apply isocliism betwee groups to obtai the relative commutativity degrees for some other classes of groups. Theorem 4.. Let G = D 2 = a, b : a = b 2 =, a b = a be the dihedral group of order 2. The { 2τ), odd, DG) = 2kτm), = 2 k m, k ad m odd. where τm) is the umber of divisors of m. Proof. It is kow that a arbitrary subgroup of G = D 2 has the form a k, a t b or a k, a l b, where t = 0,,, k ad l = 0,, 2,..., k. We proceed i two steps. First suppose that is odd. The d a k, G) = a k h G = k k + a ki ) G h a k i= = k + 2 ) k ) = 2 + k 2, d a t b, G) = a t b h G = + ) 2 h a l b d a k, a l b = a k, a l b = k + 2 = k h a k,a l b k i= h G a ki ) G + k ) + k i= = 2 + 2, a ki+l b) G )) 2 k k = k 4, where k ad l = 0,, 2,..., k. Thus DG) = { 2 + k 2, k 4 : k, k } ad a simple verificatio shows that DG) = 2τ).
8 278 Barzegar, Erfaia ad Farrokhi Now suppose that = 2 k m is eve, where k ad m is odd. The k + 2 k )) = 2 + k 2, k odd, d a k, G) = a k h a k h G = k k 2)) = 2 + k, k eve. Sice a l b) G = {a l+2i b : 0 i 2 } we have d al b, G) = 2 + ) = 2 +. Also d a k, a l b, G) = 4 + k 4 +, 4 + k 2 +, k odd, k eve. Therefore { DD 2 ) = 2 + k, 2 + k 2 2, k 3 2, k 4 4 : k, k 2, k 3, k 4,, are eve ad, } are odd, k k 3 k 2 k 4 where k 3, k 4. We cosider the followig cases: ) If 2 + k = k 3 2, the 2 = k 3+2 k ) ad k 3 +2 k ) < 0 if k >. Thus k = ad k 3 = 2. 2) If 2 + k = k 4 4, the = k 4+4 k ) ad k 4 +2 k ) < 0 if k >. Thus k = ad k 4 =, which is a cotradictio. 3) If 2 + k 2 2 = k 3 2, the 2 = k k 2 ad k k 2 < 0 if k 2 > 2. Thus k 2 2. Sice k 2 is odd we should have k 2 = 2, hece k 3 = 2. 4) If 2 + k 2 2 = k 4 4, the = k k 2 ad k k 2 < 0 if k 2 > 2. Thus k 2 2, which is a cotradictio. 5) If 2 + k = 2 + k 2 2, the k 2 = 2k. 6) If k 3 2 = k 4 4, the k 4 = 2k 3. Now, by utilizig the cases )-6), the result follows. Corollary 4.2. DD 2 ) = 3 if ad oly if = p or 2p, where p is a prime. Defiitio 4.3. Let G ad G 2 be two groups ad H ad H 2 be subgroup of G ad G 2, respectively. Suppose that α is a isomorphism from G /ZG ) to G 2 /ZG 2 ) such that its restrictio to H /H ZG ) is a isomorphism from H /H ZG ) to H 2 /H 2 ZG 2 ) ad β is a isomorphism from [H, G ] to [H 2, G 2 ]. The the pair α, β) is called a 2
9 Relative commutativity degrees 279 relative isocliism from H, G ) to H 2, G 2 ) if the followig diagram is commutative: H H ZG ) G ZG ) α 2 H 2 H 2 ZG 2 ) G 2 ZG 2 ) where ad γ [H, G ] β γ 2 [H 2, G 2 ] γ h H ZG )), g ZG )) = [h, g ] γ 2 h 2 H 2 ZG 2 )), g 2 ZG 2 )) = [h 2, g 2 ] for each h H, h 2 H 2, g G ad g 2 G 2. If H = G ad H 2 = G 2, the we say that G ad G 2 are isocliic. As a immediate cosequet of the above defiitio we have the followig result. Lemma 4.4. If G ad G 2 are two isocliic groups, the DG ) = DG 2 ). Usig Lemma 4.4 ad Theorem 4. we obtai the followig results. Note that the geeralized quarerio groups Q 4 ad quasi-dihedral groups QD 2 3) are isocliic with the groups D 4 ad D 2, respectively. Corollary 4.5. If = 2 k m m odd), the DQ 4 ) = 2k +)τm). Corollary 4.6. If 3, the DQD 2 ) = 2 3. Ackowledgmets The secod author would like to thak Ferdowsi Uiversity of Mashhad for partial support of this research Grat No. MP90257ERF). Refereces [] A. Erfaia, R. Rezaei ad P. Lescot, O the relative commutativity degree of a subgroup of a fiite group, Comm. Algebra ), o. 2, [2] A. Erfaia ad R. Rezaei, A ote o the relative commutativity degree of fiite groups, Submitted.
10 280 Barzegar, Erfaia ad Farrokhi [3] M. Farrokhi D. G., Fiite groups with two subgroup ormality degrees, Submitted. [4] W. H. Gustafso, What is the probability that two group elemets commute? Amer. Math. Mothly ) [5] P. Lescot, Isocliism classes ad commutativity degrees of fiite groups, J. Algebra ), o. 3, [6] F. Saeedi, M. Farrokhi D. G. ad S. H. Jafari, Subgroup ormality degrees of fiite groups I, Arch. Math. Basel) 96 20), o. 3, R. Barzgar Departmet of Pure Mathematics, Ferdowsi Uiversity of Mashhad, P.O. Box 59, 9775 Mashhad, Ira A. Erfaia Departmet of Pure Mathematics ad Ceter of Excellece i Aalysis o Algebraic Structures, Ferdowsi Uiversity of Mashhad, P.O. Box 59, 9775 Mashhad, Ira erfaia@math.um.ac.ir M. Farrokhi D. G. Departmet of Pure Mathematics, Ferdowsi Uiversity of Mashhad, P.O. Box 59, 9775 Mashhad, Ira m.farrokhi.d.g@gmail.com
The normal subgroup structure of ZM-groups
arxiv:1502.04776v1 [math.gr] 17 Feb 2015 The ormal subgroup structure of ZM-groups Marius Tărăuceau February 17, 2015 Abstract The mai goal of this ote is to determie ad to cout the ormal subgroups of
More informationFINITE MULTIPLICATIVE SUBGROUPS IN DIVISION RINGS
FINITE MULTIPLICATIVE SUBGROUPS IN DIVISION RINGS I. N. HERSTEIN 1. Itroductio. If G is a fiite subgroup of the multiplicative group of ozero elemets of a commutative field, the it is kow that G must be
More informationAutocommutator Subgroups of Finite Groups
JOURNAL OF ALGEBRA 90, 556562 997 ARTICLE NO. JA96692 Autocommutator Subgroups of Fiite Groups Peter V. Hegarty Departmet of Mathematics, Priceto Uiersity, Priceto, New Jersey 08544 Commuicated by Gordo
More informationThe 4-Nicol Numbers Having Five Different Prime Divisors
1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 14 (2011), Article 11.7.2 The 4-Nicol Numbers Havig Five Differet Prime Divisors Qiao-Xiao Ji ad Mi Tag 1 Departmet of Mathematics Ahui Normal Uiversity
More informationCommutativity in Permutation Groups
Commutativity i Permutatio Groups Richard Wito, PhD Abstract I the group Sym(S) of permutatios o a oempty set S, fixed poits ad trasiet poits are defied Prelimiary results o fixed ad trasiet poits are
More informationACO Comprehensive Exam 9 October 2007 Student code A. 1. Graph Theory
1. Graph Theory Prove that there exist o simple plaar triagulatio T ad two distict adjacet vertices x, y V (T ) such that x ad y are the oly vertices of T of odd degree. Do ot use the Four-Color Theorem.
More informationON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS
ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS NORBERT KAIBLINGER Abstract. Results of Lid o Lehmer s problem iclude the value of the Lehmer costat of the fiite cyclic group Z/Z, for 5 ad all odd. By complemetary
More informationSolutions to Math 347 Practice Problems for the final
Solutios to Math 347 Practice Problems for the fial 1) True or False: a) There exist itegers x,y such that 50x + 76y = 6. True: the gcd of 50 ad 76 is, ad 6 is a multiple of. b) The ifiimum of a set is
More informationChain conditions. 1. Artinian and noetherian modules. ALGBOOK CHAINS 1.1
CHAINS 1.1 Chai coditios 1. Artiia ad oetheria modules. (1.1) Defiitio. Let A be a rig ad M a A-module. The module M is oetheria if every ascedig chai!!m 1 M 2 of submodules M of M is stable, that is,
More informationOn matchings in hypergraphs
O matchigs i hypergraphs Peter Frakl Tokyo, Japa peter.frakl@gmail.com Tomasz Luczak Adam Mickiewicz Uiversity Faculty of Mathematics ad CS Pozań, Polad ad Emory Uiversity Departmet of Mathematics ad CS
More informationSOLVED EXAMPLES
Prelimiaries Chapter PELIMINAIES Cocept of Divisibility: A o-zero iteger t is said to be a divisor of a iteger s if there is a iteger u such that s tu I this case we write t s (i) 6 as ca be writte as
More informationON MEAN ERGODIC CONVERGENCE IN THE CALKIN ALGEBRAS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX0000-0 ON MEAN ERGODIC CONVERGENCE IN THE CALKIN ALGEBRAS MARCH T. BOEDIHARDJO AND WILLIAM B. JOHNSON 2
More informationChapter 0. Review of set theory. 0.1 Sets
Chapter 0 Review of set theory Set theory plays a cetral role i the theory of probability. Thus, we will ope this course with a quick review of those otios of set theory which will be used repeatedly.
More informationThe log-behavior of n p(n) and n p(n)/n
Ramauja J. 44 017, 81-99 The log-behavior of p ad p/ William Y.C. Che 1 ad Ke Y. Zheg 1 Ceter for Applied Mathematics Tiaji Uiversity Tiaji 0007, P. R. Chia Ceter for Combiatorics, LPMC Nakai Uivercity
More information1. By using truth tables prove that, for all statements P and Q, the statement
Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3
More informationSeunghee Ye Ma 8: Week 5 Oct 28
Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value
More informationChapter 2. Periodic points of toral. automorphisms. 2.1 General introduction
Chapter 2 Periodic poits of toral automorphisms 2.1 Geeral itroductio The automorphisms of the two-dimesioal torus are rich mathematical objects possessig iterestig geometric, algebraic, topological ad
More informationRegular Elements and BQ-Elements of the Semigroup (Z n, )
Iteratioal Mathematical Forum, 5, 010, o. 51, 533-539 Regular Elemets ad BQ-Elemets of the Semigroup (Z, Ng. Dapattaamogko ad Y. Kemprasit Departmet of Mathematics, Faculty of Sciece Chulalogkor Uiversity,
More informationA Note On The Exponential Of A Matrix Whose Elements Are All 1
Applied Mathematics E-Notes, 8(208), 92-99 c ISSN 607-250 Available free at mirror sites of http://wwwmaththuedutw/ ame/ A Note O The Expoetial Of A Matrix Whose Elemets Are All Reza Farhadia Received
More informationThe Structure of Z p when p is Prime
LECTURE 13 The Structure of Z p whe p is Prime Theorem 131 If p > 1 is a iteger, the the followig properties are equivalet (1) p is prime (2) For ay [0] p i Z p, the equatio X = [1] p has a solutio i Z
More informationDirichlet s Theorem on Arithmetic Progressions
Dirichlet s Theorem o Arithmetic Progressios Athoy Várilly Harvard Uiversity, Cambridge, MA 0238 Itroductio Dirichlet s theorem o arithmetic progressios is a gem of umber theory. A great part of its beauty
More informationHoggatt and King [lo] defined a complete sequence of natural numbers
REPRESENTATIONS OF N AS A SUM OF DISTINCT ELEMENTS FROM SPECIAL SEQUENCES DAVID A. KLARNER, Uiversity of Alberta, Edmoto, Caada 1. INTRODUCTION Let a, I deote a sequece of atural umbers which satisfies
More informationBertrand s Postulate
Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a
More informationExpected Norms of Zero-One Polynomials
DRAFT: Caad. Math. Bull. July 4, 08 :5 File: borwei80 pp. Page Sheet of Caad. Math. Bull. Vol. XX (Y, ZZZZ pp. 0 0 Expected Norms of Zero-Oe Polyomials Peter Borwei, Kwok-Kwog Stephe Choi, ad Idris Mercer
More informationMATH 324 Summer 2006 Elementary Number Theory Solutions to Assignment 2 Due: Thursday July 27, 2006
MATH 34 Summer 006 Elemetary Number Theory Solutios to Assigmet Due: Thursday July 7, 006 Departmet of Mathematical ad Statistical Scieces Uiversity of Alberta Questio [p 74 #6] Show that o iteger of the
More informationand each factor on the right is clearly greater than 1. which is a contradiction, so n must be prime.
MATH 324 Summer 200 Elemetary Number Theory Solutios to Assigmet 2 Due: Wedesday July 2, 200 Questio [p 74 #6] Show that o iteger of the form 3 + is a prime, other tha 2 = 3 + Solutio: If 3 + is a prime,
More informationM A T H F A L L CORRECTION. Algebra I 1 4 / 1 0 / U N I V E R S I T Y O F T O R O N T O
M A T H 2 4 0 F A L L 2 0 1 4 HOMEWORK ASSIGNMENT #4 CORRECTION Algebra I 1 4 / 1 0 / 2 0 1 4 U N I V E R S I T Y O F T O R O N T O P r o f e s s o r : D r o r B a r - N a t a Correctio Homework Assigmet
More informationDIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS
DIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS VERNER E. HOGGATT, JR. Sa Jose State Uiversity, Sa Jose, Califoria 95192 ad CALVIN T. LONG Washigto State Uiversity, Pullma, Washigto 99163
More informationIt is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function.
MATH 532 Measurable Fuctios Dr. Neal, WKU Throughout, let ( X, F, µ) be a measure space ad let (!, F, P ) deote the special case of a probability space. We shall ow begi to study real-valued fuctios defied
More informationn=1 a n is the sequence (s n ) n 1 n=1 a n converges to s. We write a n = s, n=1 n=1 a n
Series. Defiitios ad first properties A series is a ifiite sum a + a + a +..., deoted i short by a. The sequece of partial sums of the series a is the sequece s ) defied by s = a k = a +... + a,. k= Defiitio
More informationBINOMIAL PREDICTORS. + 2 j 1. Then n + 1 = The row of the binomial coefficients { ( n
BINOMIAL PREDICTORS VLADIMIR SHEVELEV arxiv:0907.3302v2 [math.nt] 22 Jul 2009 Abstract. For oegative itegers, k, cosider the set A,k = { [0, 1,..., ] : 2 k ( ). Let the biary epasio of + 1 be: + 1 = 2
More informationMath 451: Euclidean and Non-Euclidean Geometry MWF 3pm, Gasson 204 Homework 3 Solutions
Math 451: Euclidea ad No-Euclidea Geometry MWF 3pm, Gasso 204 Homework 3 Solutios Exercises from 1.4 ad 1.5 of the otes: 4.3, 4.10, 4.12, 4.14, 4.15, 5.3, 5.4, 5.5 Exercise 4.3. Explai why Hp, q) = {x
More informationThe value of Banach limits on a certain sequence of all rational numbers in the interval (0,1) Bao Qi Feng
The value of Baach limits o a certai sequece of all ratioal umbers i the iterval 0, Bao Qi Feg Departmet of Mathematical Scieces, Ket State Uiversity, Tuscarawas, 330 Uiversity Dr. NE, New Philadelphia,
More informationA TYPE OF PRIMITIVE ALGEBRA*
A TYPE OF PRIMITIVE ALGEBRA* BT J. H. M. WEDDERBURN I a recet paper,t L. E. Dickso has discussed the liear associative algebra, A, defied by the relatios xy = yo(x), y = g, where 8 ( x ) is a polyomial
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationDisjoint unions of complete graphs characterized by their Laplacian spectrum
Electroic Joural of Liear Algebra Volume 18 Volume 18 (009) Article 56 009 Disjoit uios of complete graphs characterized by their Laplacia spectrum Romai Boulet boulet@uiv-tlse.fr Follow this ad additioal
More informationMath F215: Induction April 7, 2013
Math F25: Iductio April 7, 203 Iductio is used to prove that a collectio of statemets P(k) depedig o k N are all true. A statemet is simply a mathematical phrase that must be either true or false. Here
More informationWeakly Connected Closed Geodetic Numbers of Graphs
Iteratioal Joural of Mathematical Aalysis Vol 10, 016, o 6, 57-70 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/101988/ijma01651193 Weakly Coected Closed Geodetic Numbers of Graphs Rachel M Pataga 1, Imelda
More information1 lim. f(x) sin(nx)dx = 0. n sin(nx)dx
Problem A. Calculate ta(.) to 4 decimal places. Solutio: The power series for si(x)/ cos(x) is x + x 3 /3 + (2/5)x 5 +. Puttig x =. gives ta(.) =.3. Problem 2A. Let f : R R be a cotiuous fuctio. Show that
More informationSome p-adic congruences for p q -Catalan numbers
Some p-adic cogrueces for p q -Catala umbers Floria Luca Istituto de Matemáticas Uiversidad Nacioal Autóoma de México C.P. 58089, Morelia, Michoacá, México fluca@matmor.uam.mx Paul Thomas Youg Departmet
More informationarxiv: v1 [math.nt] 10 Dec 2014
A DIGITAL BINOMIAL THEOREM HIEU D. NGUYEN arxiv:42.38v [math.nt] 0 Dec 204 Abstract. We preset a triagle of coectios betwee the Sierpisi triagle, the sum-of-digits fuctio, ad the Biomial Theorem via a
More informationLecture Notes for Analysis Class
Lecture Notes for Aalysis Class Topological Spaces A topology for a set X is a collectio T of subsets of X such that: (a) X ad the empty set are i T (b) Uios of elemets of T are i T (c) Fiite itersectios
More informationk-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c 1. Introduction
Acta Math. Uiv. Comeiaae Vol. LXXXVI, 2 (2017), pp. 279 286 279 k-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c N. IRMAK ad M. ALP Abstract. The k-geeralized Fiboacci sequece { F (k)
More informationHomework 2 January 19, 2006 Math 522. Direction: This homework is due on January 26, In order to receive full credit, answer
Homework 2 Jauary 9, 26 Math 522 Directio: This homework is due o Jauary 26, 26. I order to receive full credit, aswer each problem completely ad must show all work.. What is the set of the uits (that
More informationSOME FINITE SIMPLE GROUPS OF LIE TYPE C n ( q) ARE UNIQUELY DETERMINED BY THEIR ELEMENT ORDERS AND THEIR ORDER
Joural of Algebra, Nuber Theory: Advaces ad Applicatios Volue, Nuber, 010, Pages 57-69 SOME FINITE SIMPLE GROUPS OF LIE TYPE C ( q) ARE UNIQUELY DETERMINED BY THEIR ELEMENT ORDERS AND THEIR ORDER School
More informationAMS Mathematics Subject Classification : 40A05, 40A99, 42A10. Key words and phrases : Harmonic series, Fourier series. 1.
J. Appl. Math. & Computig Vol. x 00y), No. z, pp. A RECURSION FOR ALERNAING HARMONIC SERIES ÁRPÁD BÉNYI Abstract. We preset a coveiet recursive formula for the sums of alteratig harmoic series of odd order.
More informationName of the Student:
SUBJECT NAME : Discrete Mathematics SUBJECT CODE : MA 65 MATERIAL NAME : Problem Material MATERIAL CODE : JM08ADM010 (Sca the above QR code for the direct dowload of this material) Name of the Studet:
More informationProof of Fermat s Last Theorem by Algebra Identities and Linear Algebra
Proof of Fermat s Last Theorem by Algebra Idetities ad Liear Algebra Javad Babaee Ragai Youg Researchers ad Elite Club, Qaemshahr Brach, Islamic Azad Uiversity, Qaemshahr, Ira Departmet of Civil Egieerig,
More informationThe multiplicative structure of finite field and a construction of LRC
IERG6120 Codig for Distributed Storage Systems Lecture 8-06/10/2016 The multiplicative structure of fiite field ad a costructio of LRC Lecturer: Keeth Shum Scribe: Zhouyi Hu Notatios: We use the otatio
More informationFermat s Little Theorem. mod 13 = 0, = }{{} mod 13 = 0. = a a a }{{} mod 13 = a 12 mod 13 = 1, mod 13 = a 13 mod 13 = a.
Departmet of Mathematical Scieces Istructor: Daiva Puciskaite Discrete Mathematics Fermat s Little Theorem 43.. For all a Z 3, calculate a 2 ad a 3. Case a = 0. 0 0 2-times Case a 0. 0 0 3-times a a 2-times
More informationM17 MAT25-21 HOMEWORK 5 SOLUTIONS
M17 MAT5-1 HOMEWORK 5 SOLUTIONS 1. To Had I Cauchy Codesatio Test. Exercise 1: Applicatio of the Cauchy Codesatio Test Use the Cauchy Codesatio Test to prove that 1 diverges. Solutio 1. Give the series
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationExercises 1 Sets and functions
Exercises 1 Sets ad fuctios HU Wei September 6, 018 1 Basics Set theory ca be made much more rigorous ad built upo a set of Axioms. But we will cover oly some heuristic ideas. For those iterested studets,
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More informationAssignment 5: Solutions
McGill Uiversity Departmet of Mathematics ad Statistics MATH 54 Aalysis, Fall 05 Assigmet 5: Solutios. Let y be a ubouded sequece of positive umbers satisfyig y + > y for all N. Let x be aother sequece
More informationTHE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS
THE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS DEMETRES CHRISTOFIDES Abstract. Cosider a ivertible matrix over some field. The Gauss-Jorda elimiatio reduces this matrix to the idetity
More informationCSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
More informationMeasure and Measurable Functions
3 Measure ad Measurable Fuctios 3.1 Measure o a Arbitrary σ-algebra Recall from Chapter 2 that the set M of all Lebesgue measurable sets has the followig properties: R M, E M implies E c M, E M for N implies
More informationUniversity of Twente The Netherlands
Faculty of Mathematical Scieces t Uiversity of Twete The Netherlads P.O. Box 7 7500 AE Eschede The Netherlads Phoe: +3-53-4893400 Fax: +3-53-48934 Email: memo@math.utwete.l www.math.utwete.l/publicatios
More informationPRELIM PROBLEM SOLUTIONS
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems
More informationUniversity of Colorado Denver Dept. Math. & Stat. Sciences Applied Analysis Preliminary Exam 13 January 2012, 10:00 am 2:00 pm. Good luck!
Uiversity of Colorado Dever Dept. Math. & Stat. Scieces Applied Aalysis Prelimiary Exam 13 Jauary 01, 10:00 am :00 pm Name: The proctor will let you read the followig coditios before the exam begis, ad
More informationMATH10212 Linear Algebra B Proof Problems
MATH22 Liear Algebra Proof Problems 5 Jue 26 Each problem requests a proof of a simple statemet Problems placed lower i the list may use the results of previous oes Matrices ermiats If a b R the matrix
More informationDisjoint Systems. Abstract
Disjoit Systems Noga Alo ad Bey Sudaov Departmet of Mathematics Raymod ad Beverly Sacler Faculty of Exact Scieces Tel Aviv Uiversity, Tel Aviv, Israel Abstract A disjoit system of type (,,, ) is a collectio
More information1 Elliptic Curves Over Finite Fields
1 Elliptic Curves Over Fiite Fields 1.1 Itroductio Defiitio 1.1. Elliptic curves ca be defied over ay field K; the formal defiitio of a elliptic curve is a osigular (o cusps, self-itersectios, or isolated
More informationSquare-Congruence Modulo n
Square-Cogruece Modulo Abstract This paper is a ivestigatio of a equivalece relatio o the itegers that was itroduced as a exercise i our Discrete Math class. Part I - Itro Defiitio Two itegers are Square-Cogruet
More informationLONG SNAKES IN POWERS OF THE COMPLETE GRAPH WITH AN ODD NUMBER OF VERTICES
J Lodo Math Soc (2 50, (1994, 465 476 LONG SNAKES IN POWERS OF THE COMPLETE GRAPH WITH AN ODD NUMBER OF VERTICES Jerzy Wojciechowski Abstract I [5] Abbott ad Katchalski ask if there exists a costat c >
More informationSEMIGROUPS OF VALUATIONS DOMINATING LOCAL DOMAINS
SEMIGROUPS OF VALUATIONS DOMINATING LOCAL DOMAINS STEVEN DALE CUTKOSKY Let (R, m R ) be a equicharacteristic local domai, with quotiet field K. Suppose that ν is a valuatio of K with valuatio rig (V, m
More informationROTATION-EQUIVALENCE CLASSES OF BINARY VECTORS. 1. Introduction
t m Mathematical Publicatios DOI: 10.1515/tmmp-2016-0033 Tatra Mt. Math. Publ. 67 (2016, 93 98 ROTATION-EQUIVALENCE CLASSES OF BINARY VECTORS Otokar Grošek Viliam Hromada ABSTRACT. I this paper we study
More informationBoundaries and the James theorem
Boudaries ad the James theorem L. Vesely 1. Itroductio The followig theorem is importat ad well kow. All spaces cosidered here are real ormed or Baach spaces. Give a ormed space X, we deote by B X ad S
More informationarxiv: v1 [math.fa] 3 Apr 2016
Aticommutator Norm Formula for Proectio Operators arxiv:164.699v1 math.fa] 3 Apr 16 Sam Walters Uiversity of Norther British Columbia ABSTRACT. We prove that for ay two proectio operators f, g o Hilbert
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationBeurling Integers: Part 2
Beurlig Itegers: Part 2 Isomorphisms Devi Platt July 11, 2015 1 Prime Factorizatio Sequeces I the last article we itroduced the Beurlig geeralized itegers, which ca be represeted as a sequece of real umbers
More informationA Note on the Symmetric Powers of the Standard Representation of S n
A Note o the Symmetric Powers of the Stadard Represetatio of S David Savitt 1 Departmet of Mathematics, Harvard Uiversity Cambridge, MA 0138, USA dsavitt@mathharvardedu Richard P Staley Departmet of Mathematics,
More informationSection 5.1 The Basics of Counting
1 Sectio 5.1 The Basics of Coutig Combiatorics, the study of arragemets of objects, is a importat part of discrete mathematics. I this chapter, we will lear basic techiques of coutig which has a lot of
More informationMath 4400/6400 Homework #7 solutions
MATH 4400 problems. Math 4400/6400 Homewor #7 solutios 1. Let p be a prime umber. Show that the order of 1 + p modulo p 2 is exactly p. Hit: Expad (1 + p) p by the biomial theorem, ad recall from MATH
More informationCombinatorics and Newton s theorem
INTRODUCTION TO MATHEMATICAL REASONING Key Ideas Worksheet 5 Combiatorics ad Newto s theorem This week we are goig to explore Newto s biomial expasio theorem. This is a very useful tool i aalysis, but
More informationAn elementary proof that almost all real numbers are normal
Acta Uiv. Sapietiae, Mathematica, 2, (200 99 0 A elemetary proof that almost all real umbers are ormal Ferdiád Filip Departmet of Mathematics, Faculty of Educatio, J. Selye Uiversity, Rolícej šoly 59,
More information# fixed points of g. Tree to string. Repeatedly select the leaf with the smallest label, write down the label of its neighbour and remove the leaf.
Combiatorics Graph Theory Coutig labelled ad ulabelled graphs There are 2 ( 2) labelled graphs of order. The ulabelled graphs of order correspod to orbits of the actio of S o the set of labelled graphs.
More informationA 2nTH ORDER LINEAR DIFFERENCE EQUATION
A 2TH ORDER LINEAR DIFFERENCE EQUATION Doug Aderso Departmet of Mathematics ad Computer Sciece, Cocordia College Moorhead, MN 56562, USA ABSTRACT: We give a formulatio of geeralized zeros ad (, )-discojugacy
More informationBertrand s Postulate. Theorem (Bertrand s Postulate): For every positive integer n, there is a prime p satisfying n < p 2n.
Bertrad s Postulate Our goal is to prove the followig Theorem Bertrad s Postulate: For every positive iteger, there is a prime p satisfyig < p We remark that Bertrad s Postulate is true by ispectio for,,
More information11. FINITE FIELDS. Example 1: The following tables define addition and multiplication for a field of order 4.
11. FINITE FIELDS 11.1. A Field With 4 Elemets Probably the oly fiite fields which you ll kow about at this stage are the fields of itegers modulo a prime p, deoted by Z p. But there are others. Now although
More informationFactors of sums and alternating sums involving binomial coefficients and powers of integers
Factors of sums ad alteratig sums ivolvig biomial coefficiets ad powers of itegers Victor J. W. Guo 1 ad Jiag Zeg 2 1 Departmet of Mathematics East Chia Normal Uiversity Shaghai 200062 People s Republic
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationModern Algebra. Previous year Questions from 2017 to Ramanasri
Moder Algebra Previous year Questios from 017 to 199 Ramaasri 017 S H O P NO- 4, 1 S T F L O O R, N E A R R A P I D F L O U R M I L L S, O L D R A J E N D E R N A G A R, N E W D E L H I. W E B S I T E
More informationPairs of disjoint q-element subsets far from each other
Pairs of disjoit q-elemet subsets far from each other Hikoe Eomoto Departmet of Mathematics, Keio Uiversity 3-14-1 Hiyoshi, Kohoku-Ku, Yokohama, 223 Japa, eomoto@math.keio.ac.jp Gyula O.H. Katoa Alfréd
More informationON STATISTICAL CONVERGENCE AND STATISTICAL MONOTONICITY
Aales Uiv. Sci. Budapest., Sect. Comp. 39 (203) 257 270 ON STATISTICAL CONVERGENCE AND STATISTICAL MONOTONICITY E. Kaya (Mersi, Turkey) M. Kucukasla (Mersi, Turkey) R. Wager (Paderbor, Germay) Dedicated
More informationON SOME DIOPHANTINE EQUATIONS RELATED TO SQUARE TRIANGULAR AND BALANCING NUMBERS
Joural of Algebra, Number Theory: Advaces ad Applicatios Volume, Number, 00, Pages 7-89 ON SOME DIOPHANTINE EQUATIONS RELATED TO SQUARE TRIANGULAR AND BALANCING NUMBERS OLCAY KARAATLI ad REFİK KESKİN Departmet
More informationKU Leuven Department of Computer Science
O orthogoal polyomials related to arithmetic ad harmoic sequeces Adhemar Bultheel ad Adreas Lasarow Report TW 687, February 208 KU Leuve Departmet of Computer Sciece Celestijelaa 200A B-300 Heverlee (Belgium)
More informationc 2006 Society for Industrial and Applied Mathematics
SIAM J. MATRIX ANAL. APPL. Vol. 7, No. 3, pp. 851 860 c 006 Society for Idustrial ad Applied Mathematics EXTREMAL EIGENVALUES OF REAL SYMMETRIC MATRICES WITH ENTRIES IN AN INTERVAL XINGZHI ZHAN Abstract.
More informationJurnal Teknologi THE SQUARED COMMUTATIVITY DEGREE OF DIHEDRAL GROUPS. Full Paper
Jural Tekologi THE SQUARE COMMUTATIVITY EREE OF IHERAL ROUPS Muhaizah Abdul Hamid a, Nor Muhaiiah Mohd Ali a*, Nor Haiza Sarmi a, Ahmad Erfaia b, Fadila Normahia Abd Maaf a a epartmet of Mathematical Scieces,
More informationMATH 112: HOMEWORK 6 SOLUTIONS. Problem 1: Rudin, Chapter 3, Problem s k < s k < 2 + s k+1
MATH 2: HOMEWORK 6 SOLUTIONS CA PRO JIRADILOK Problem. If s = 2, ad Problem : Rudi, Chapter 3, Problem 3. s + = 2 + s ( =, 2, 3,... ), prove that {s } coverges, ad that s < 2 for =, 2, 3,.... Proof. The
More informationSOME TRIBONACCI IDENTITIES
Mathematics Today Vol.7(Dec-011) 1-9 ISSN 0976-38 Abstract: SOME TRIBONACCI IDENTITIES Shah Devbhadra V. Sir P.T.Sarvajaik College of Sciece, Athwalies, Surat 395001. e-mail : drdvshah@yahoo.com The sequece
More informationStudy of Pseudo BL Algebras in View of Left Boolean Lifting Property
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 13, Issue 1 (Jue 2018), pp. 354-381 Applicatios ad Applied Mathematics: A Iteratioal Joural (AAM) Study of Pseudo BL Algebras i
More informationOn forward improvement iteration for stopping problems
O forward improvemet iteratio for stoppig problems Mathematical Istitute, Uiversity of Kiel, Ludewig-Mey-Str. 4, D-24098 Kiel, Germay irle@math.ui-iel.de Albrecht Irle Abstract. We cosider the optimal
More informationLecture 4: Grassmannians, Finite and Affine Morphisms
18.725 Algebraic Geometry I Lecture 4 Lecture 4: Grassmaias, Fiite ad Affie Morphisms Remarks o last time 1. Last time, we proved the Noether ormalizatio lemma: If A is a fiitely geerated k-algebra, the,
More informationOn Some Properties of Digital Roots
Advaces i Pure Mathematics, 04, 4, 95-30 Published Olie Jue 04 i SciRes. http://www.scirp.org/joural/apm http://dx.doi.org/0.436/apm.04.46039 O Some Properties of Digital Roots Ilha M. Izmirli Departmet
More informationGROUPS AND APPLICATIONS
MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Groups ad Applicatios - Tadao ODA GROUPS AND APPLICATIONS Tadao ODA Tohoku Uiversity, Japa Keywords: group, homomorphism, quotiet group, group actio, trasformatio,
More informationA NOTE ON SPECTRAL CONTINUITY. In Ho Jeon and In Hyoun Kim
Korea J. Math. 23 (2015), No. 4, pp. 601 605 http://dx.doi.org/10.11568/kjm.2015.23.4.601 A NOTE ON SPECTRAL CONTINUITY I Ho Jeo ad I Hyou Kim Abstract. I the preset ote, provided T L (H ) is biquasitriagular
More informationMath 2112 Solutions Assignment 5
Math 2112 Solutios Assigmet 5 5.1.1 Idicate which of the followig relatioships are true ad which are false: a. Z Q b. R Q c. Q Z d. Z Z Z e. Q R Q f. Q Z Q g. Z R Z h. Z Q Z a. True. Every positive iteger
More information