On intransitive graph-restrictive permutation groups
|
|
- Maximillian Bryan
- 5 years ago
- Views:
Transcription
1 J Algebr Comb (2014) 40: DOI /s On ntranstve graph-restrctve permutaton groups Pablo Spga Gabrel Verret Receved: 5 December 2012 / Accepted: 5 October 2013 / Publshed onlne: 26 October 2013 Sprnger Scence+Busness Meda New York 2013 Abstract Let Γ be a fnte connected G-vertex-transtve graph and let v be a vertex of Γ If the permutaton group nduced by the acton of the vertex-stablser G v on the neghbourhood Γ(v) s permutaton somorphc to L, then (Γ, G) s sad to be locally L A permutaton group L s graph-restrctve f there exsts a constant c(l) such that, for every locally L par (Γ, G) and a vertex v of Γ, the nequalty G v c(l) holds We show that an ntranstve group s graph-restrctve f and only f t s semregular Keywords Vertex-transtve Graph-restrctve Semregular 1 Introducton A graph Γ s sad to be G-vertex-transtve f G s a subgroup of Aut(Γ ) actng transtvely on the vertex set of Γ LetΓ be a fnte, connected, smple G-vertex-transtve graph and let v be a vertex of Γ If the permutaton group nduced by the acton of the vertex-stablser G v on the neghbourhood Γ(v)s permutaton somorphc to L, then (Γ, G) s sad to be locally L Note that, up to permutaton somorphsm, L does not depend on the choce of v, and, moreover, the degree of L s equal to the valency of Γ In[6, p 499], the second author ntroduced the followng defnton P Spga Departmento d Matematca Pura e Applcata, Unversty of Mlano-Bcocca, Va Cozz 53, Mlano, Italy e-mal: pablospga@unmbt G Verret (B) Faculty of Mathematcs, Natural Scences and Info Tech, Unversty of Prmorska, Glagoljaška 8, 6000 Koper, Slovena e-mal: gabrelverret@fmfun-ljs
2 180 J Algebr Comb (2014) 40: Defnton 11 A permutaton group L s graph-restrctve f there exsts a constant c(l) such that, for every locally L par (Γ, G) and for every vertex v of Γ, the nequalty G v c(l) holds To be precse, Defnton 11 s a generalsaton of the defnton from [6], where the group L s assumed to be transtve The problem of determnng whch transtve permutaton groups are graph-restrctve was also proposed n [6] A survey of the state of ths problem can be found n [3], where t was conjectured ([3, Conjecture 3]) that a transtve permutaton group s graph-restrctve f and only f t s semprmtve (A permutaton group s sad to be semregular f each of ts pont-stablsers s trval and semprmtve f each of ts normal subgroups s ether transtve or semregular) Havng removed the requrement of transtvty from the defnton of graphrestrctve, t s then natural to try to determne whch ntranstve permutaton groups are graph-restrctve The man result of ths note s a complete soluton to ths problem (whch we dd not expect, gven the abundance and relatve lack of structure of ntranstve groups) Theorem 12 An ntranstve and graph-restrctve permutaton group s semregular It s easly seen that a semregular permutaton group s graph-restrctve Indeed, f L s a semregular permutaton group of degree d and (Γ, G) s locally L, then for every arc vw of Γ the group G vw fxes the neghbourhood Γ(v) pontwse Snce Γ s connected, t follows that G vw = 1 and hence G v Γ(v) =d and L s graph-restrctve Thus Theorem 12 provdes a charactersaton of ntranstve graph-restrctve groups Corollary 13 An ntranstve permutaton group s graph-restrctve f and only f t s semregular Note that an ntranstve permutaton group s semregular f and only f t s semprmtve In partcular, Corollary 13 completely settles the ntranstve verson of [3, Conjecture 3], gvng remarkable new evdence towards ts veracty 2 Proof of Theorem 12 For the remander of ths paper, let L be a permutaton group on a fnte set Ω whch s nether transtve nor semregular We show that L s not graph-restrctve, from whch Theorem 12 follows 21 The constructon Let ω 1,,ω k Ω be a set of representatves of the orbts of L on Ω Snce L s not transtve, k 2 and, snce L s not semregular, we may assume wthout loss of
3 J Algebr Comb (2014) 40: Fg 1 Subgroup lattce for Sect 21 generalty that L ω1 1 Let n 2 be an nteger and let b 1 be the automorphsm of L ω1 L n ω 1 = L n+1 ω 1 defned by (x 0,x 1,,x n 1,x n ) b 1 = (x n,x n 1,,x 1,x 0 ), for each (x 0,,x n ) L n+1 ω 1 Smlarly, let b 2 be the automorphsm of L n ω 1 defned by (x 1,x 2,,x n 1,x n ) b 2 = (x n,x n 1,,x 2,x 1 ), for each (x 1,,x n ) L n ω 1 Clearly, b 1 and b 2 are nvolutons, that s, b 2 1 = 1 and b 2 2 = 1 Now, let b 3,, b k be cyclc groups of order 2 and consder the followng abstract groups: A := L L n ω 1, B 1 := ( L ω1 L n ω 1 ) b1, B 2 := L ω2 ( L n ω 1 b 2 ), B := L ω L n ω 1 b, C := L ω L n ω 1, for {3,,k}, for {1,,k}, where b 1,,b k / A For every {1,,k}, there s an obvous embeddng of C n both A and B Hence, n what follows, we regard C as a subgroup of both A and B Note that, for each {1,,k}, wehavea B = C, B : C =2 and A : C = L : L ω (see Fg 1) Lemma 21 The core of C 1 C k n A s 1 L n ω 1 Proof Let K be the core of C 1 C k n A Then K = (C 1 C k ) a = (Lω1 L ωk ) L a A a A( n ) a ω 1 Recall that L s a permutaton group on Ω and that ω 1,,ω k are representatves of the orbts of L on Ω We thus fnd that L ω1 L ωk s core-free n L and hence K = 1 L n ω 1
4 182 J Algebr Comb (2014) 40: Fg 2 The graph of groups Y Let T be the group gven by generators and relators T := A,B 1,,B k R, where R conssts only of the relatons n A,B 1,,B k together wth the dentfcaton of C n A and B, for every {1,,k} We wll obtan some basc propertes of T, whch can be deduced from any textbook on groups actng on graphs, such as [1, 2, 5] We have adopted the notaton and termnology of [2] and wll follow closely [2, I4] Usng ths termnology, the group T s exactly the fundamental group of the graph of groups Y shownnfg2 The vertces of Y are A, B 1,,B k and, for each {1,,k}, there s a (drected) edge C from A to B It follows from [2, I46] that the mages of A,B 1,,B k,c 1,,C k n T are somorphc to A,B 1,,B k,c 1,,C k, respectvely Ths allows us to dentfy A,B 1,,B k,c 1,,C k wth ther somorphc mages n T n what follows In partcular, for each {1,,k} we stll have the equaltes A B = C, B : C =2 and A : C = L : L ω n T LetT be the graph wth vertex set V T = T/A T/B 1 T/B k, (where denotes the dsjont unon) and edge-set ET = { {Ax, B x} x T, {1,k} } 22 Results about the group T and the graph T Clearly, the acton of T by rght multplcaton on V T nduces a group of automorphsms of T Under ths acton, the group T has exactly k + 1 orbts on V T, namely T/A, T/B 1,,T/B k, and k orbts on ET wth representatves {A,B 1 },,{A,B k } Ths nduces a (k + 1)-partton of the graph T Observe that the set of neghbours of A n T/B s {B a a A} As A : (A B ) = A: C = L : L w, we see that A has L : L w neghbours n T/B It follows that A has valency k =1 L : L ω = Ω A symmetrc argument, wth the roles of A and B reversed, shows that B has valency B : C =2 In partcular, T s a (2, Ω )-regular graph Lemma 22 The stablser of the vertex A n T s the subgroup A and the kernel of the acton on the neghbourhood of A s 1 L n ω 1 Proof The defnton of T mmedately shows that A s the stablser n T of the vertex A Moreover, the neghbourhood of A s T (A) ={B a {1,,k},a A}
5 J Algebr Comb (2014) 40: Fg 3 Illustraton for the proof of Lemma 23 Let K be the kernel of the acton of A on T (A) and let x K Clearly, B ax = B a f and only f axa 1 B, that s, axa 1 A B = C It follows by Lemma 21 that K = 1 L n ω 1 One of the most mportant and fundamental propertes of T s that t s a tree (see [2, I44]) We now deduce some consequences from ths pvotal result Lemma 23 For each {1,,k}, we have A A b = C Proof We argue by contradcton and assume that A A b C for some {1,,k} As B : C =2, we see that B normalses C and hence C <A A b In partcular, there exst a,a A \ C wth a = a b = b 1 ab It follows that A, B, Ab and B ab are dstnct vertces of T Now, the defnton of T shows that (A, B,Ab,B ab,ab 1 ab = A) s a cycle of length 4 n T (see Fg 3) Ths contradcts the fact that T s a tree and concludes the proof Lemma 24 The subgroup A s core-free n T In partcular, the group T acts fathfully on T/A Proof Let N be the core of A n T From Lemma 23, we obtan N C 1 C k, and t follows from Lemma 21 that N 1 L n ω 1 By constructon, the group b 1,b 2 nduces a transtve permutaton group on the n + 1 coordnates of L ω1 L n ω 1 As the frst coordnate of the elements of N s 1 and as N s nvarant under b 1,b 2,we see that every coordnate of N must be equal to 1, that s, N = 1 The lemma now follows As every vertex of T not n T/A has valency 2, we see that T s the subdvson graph of a tree T 0 wth vertex set T/A and valency Ω Clearly, T acts transtvely and, n vew of Lemma 24, fathfully on the vertces of T 0 The tree T 0 and the group T are our man ngredents for the proof of Theorem 12 (The auxlary graph T was ntroduced manly to make t more convenent to apply the results from [2]) Lemma 25 The stablser n T of the vertex A of T 0 s the subgroup A and the acton nduced by A on ts neghbourhood s permutaton somorphc to the acton of L on Ω Proof Let π : A L be the natural projecton onto the frst coordnate In other words, f a = (a 0,a 1,,a n ) A, wth a 0 L and wth a 1,,a n L ω1, then π(a) = a 0 Clearly, the kernel of π s 1 L n ω 1, whch by Lemma 22 s also the kernel of the acton of A on the neghbourhood of the vertex A Denote by T 0 (A) the neghbourhood of A n T 0 The defntons of T and T 0 yeld T 0 (A) ={Ab a
6 184 J Algebr Comb (2014) 40: {1,,k},a A} Letϕ : T 0 (A) Ω be the mappng ϕ : Ab a ω π(a) Weshow that ϕ s well-defned and njectve Indeed, Ab a = Ab a for some a,a A f and only f Ab a(a ) 1 b 1 = A, that s, a(a ) 1 A A b By Lemma 23, A A b = C Clearly, a(a ) 1 C f and only f π(a(a ) 1 ) L ω, that s, ω π(a) = ω π(a ) Ths shows that ϕ s well-defned and that t s njectve Clearly, ϕ s surjectve and hence t s a bjecton For every a,x A and for every {1,,k}, wehaveϕ((ab a)x) = (ϕ(ab a)) π(x) Asϕ s a bjecton, ths shows that the acton of A on T 0 (A) s permutaton somorphc to the acton of L on Ω Recall that a group X s sad to be resdually fnte f there exsts a famly {X m } m N of normal subgroups of fnte ndex n X wth m N X m = 1 Lemma 26 The group T s resdually fnte Proof As the groups A, B 1,,B k are fnte, t follows from [2, I47] that there exst a fnte group F and a group homomorphsm π : T F wth Ker π A = 1 and Ker π B = 1 for each {1,,k} Wrte K = Ker π Snce F s fnte, we have T : K < Snce K T, K A = 1 and K B = 1, t follows that the only element of K fxng a vertex of T s 1 and hence, by [2, I54], K s a free group In partcular, K s resdually fnte (see [4, 619] for example) It follows that there exsts a famly {K m } m N of normal subgroups of fnte ndex n K wth m N K m = 1 Let T m be the core of K m n T As T : K m = T : K K : K m <, wesee that T : T m < Moreover, snce T m K m,wehave m N T m = 1 and the lemma follows 23 Proof of Theorem 12 We now recall the defnton of a normal quotent of a graph Let Γ be a G-vertextranstve graph and let N be a normal subgroup of G Letv N denote the N-orbt contanng v VΓ Then the normal quotent Γ/N s the graph whose vertces are the N-orbts on VΓ, wth an edge between dstnct vertces v N and w N f and only f there s an edge {v,w } of Γ for some v v N and some w w N Observe that the group G/N acts transtvely on the graph Γ/N Lemma 27 There exsts a locally L par (Γ n,g n ) such that the stablser of a vertex of Γ n n G n has order L L ω1 n Proof By Lemma 26, T s resdually fnte and hence there exsts a famly {T m } m N of normal subgroups of fnte ndex n T wth m N T m = 1 Consder the set X = { a 1 b 1 a 2 b j a 3 a 1,a 2,a 3 A,,j {1,,k} } Observe that snce A s fnte, so s X In partcular, as m N T m = 1 and 1 X, there exsts m N wth X T m = 1 Let G n = T/T m and Γ n = T 0 /T m As T : T m <,
7 J Algebr Comb (2014) 40: the group G n and the graph Γ n are fnte Note that Γ n s connected and G n -vertextranstve We frst show that Γ n has valency Ω We argue by contradcton and suppose that Γ n has valency less than Ω Itfollows from the defnton of normal quotent that the vertex A of T 0 must have two dstnct neghbours n the same T m -orbt Recall that the neghbourhood of A n T 0 s {Ab a {1,,k},a A} In partcular, Ab a Ab j a and Ab an = Ab j a,for some, j {1,,k}, a,a A and n T m It follows that n a 1 b 1 Ab j a X and hence n X T m = 1, whch s a contradcton Let K be the kernel of the acton of G n on VΓ n Snce the valency of Γ n equals the valency of T 0, we see that Γ n s a regular cover of T 0 Snce Γ n s connected, t follows that K acts semregularly on VΓ n and hence K = T m By Lemma 25, (T 0,T) s locally L and hence so s (Γ n,g n ) Fnally, the stablser of the vertex AT m of Γ n s AT m /T m = A/(A Tm ) = A, whch has order A = L L ω1 n Proof of Theorem 12 By Lemma 27, for every natural nteger n 2, there exsts a locally L par (Γ n,g n ) wth (G n ) v = L L ω1 n,forv VΓ n As L ω1 > 1, ths shows that L s not graph-restrctve References 1 Dcks, W, Dunwoody, MJ: Groups Actngs on Graphs Cambrdge Studes n Advanced Mathematcs, vol 17 Cambrdge Unversty Press, Cambrdge (1989) 2 Goldschmdt, D: Graphs and groups In: Delgado, A, Goldschmdt, D, Stellmacher, B (eds) Groups and Graphs: New Results and Methods DMV Semnar, vol 6 Brkhäuser, Basel (1985) 3 Potočnk, P, Spga, P, Verret, G: On graph-restrctve permutaton groups J Comb Theory, Ser B 102, (2012) 4 Robnson, DJS: A Course n the Theory of Groups Sprnger, New York (1980) 5 Serre, JP: Trees Sprnger, New York (1980) 6 Verret, G: On the order of arc-stablzers n arc-transtve graphs Bull Aust Math Soc 80, (2009)
Graph Reconstruction by Permutations
Graph Reconstructon by Permutatons Perre Ille and Wllam Kocay* Insttut de Mathémathques de Lumny CNRS UMR 6206 163 avenue de Lumny, Case 907 13288 Marselle Cedex 9, France e-mal: lle@ml.unv-mrs.fr Computer
More informationThe probability that a pair of group elements is autoconjugate
Proc. Indan Acad. Sc. (Math. Sc.) Vol. 126, No. 1, February 2016, pp. 61 68. c Indan Academy of Scences The probablty that a par of group elements s autoconjugate MOHAMMAD REZA R MOGHADDAM 1,2,, ESMAT
More informationProblem Do any of the following determine homomorphisms from GL n (C) to GL n (C)?
Homework 8 solutons. Problem 16.1. Whch of the followng defne homomomorphsms from C\{0} to C\{0}? Answer. a) f 1 : z z Yes, f 1 s a homomorphsm. We have that z s the complex conjugate of z. If z 1,z 2
More informationDigraph representations of 2-closed permutation groups with a normal regular cyclic subgroup
Dgraph representatons of 2-closed permutaton groups wth a normal regular cyclc subgroup Jng Xu Department of Mathematcs Captal Normal Unversty Bejng 100048, Chna xujng@cnu.edu.cn Submtted: Mar 30, 2015;
More informationMath 594. Solutions 1
Math 594. Solutons 1 1. Let V and W be fnte-dmensonal vector spaces over a feld F. Let G = GL(V ) and H = GL(W ) be the assocated general lnear groups. Let X denote the vector space Hom F (V, W ) of lnear
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationSUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION
talan journal of pure appled mathematcs n. 33 2014 (63 70) 63 SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION M.R. Farhangdoost Department of Mathematcs College of Scences Shraz Unversty Shraz, 71457-44776
More informationSelf-complementing permutations of k-uniform hypergraphs
Dscrete Mathematcs Theoretcal Computer Scence DMTCS vol. 11:1, 2009, 117 124 Self-complementng permutatons of k-unform hypergraphs Artur Szymańsk A. Paweł Wojda Faculty of Appled Mathematcs, AGH Unversty
More informationA CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS
Journal of Mathematcal Scences: Advances and Applcatons Volume 25, 2014, Pages 1-12 A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS JIA JI, WEN ZHANG and XIAOFEI QI Department of Mathematcs
More informationMaximizing the number of nonnegative subsets
Maxmzng the number of nonnegatve subsets Noga Alon Hao Huang December 1, 213 Abstract Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what s the maxmum
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationModulo Magic Labeling in Digraphs
Gen. Math. Notes, Vol. 7, No., August, 03, pp. 5- ISSN 9-784; Copyrght ICSRS Publcaton, 03 www.-csrs.org Avalable free onlne at http://www.geman.n Modulo Magc Labelng n Dgraphs L. Shobana and J. Baskar
More informationAn Introduction to Morita Theory
An Introducton to Morta Theory Matt Booth October 2015 Nov. 2017: made a few revsons. Thanks to Nng Shan for catchng a typo. My man reference for these notes was Chapter II of Bass s book Algebrac K-Theory
More informationFINITELY-GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN
FINITELY-GENERTED MODULES OVER PRINCIPL IDEL DOMIN EMMNUEL KOWLSKI Throughout ths note, s a prncpal deal doman. We recall the classfcaton theorem: Theorem 1. Let M be a fntely-generated -module. (1) There
More informationSL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
More informationFixed points of IA-endomorphisms of a free metabelian Lie algebra
Proc. Indan Acad. Sc. (Math. Sc.) Vol. 121, No. 4, November 2011, pp. 405 416. c Indan Academy of Scences Fxed ponts of IA-endomorphsms of a free metabelan Le algebra NAIME EKICI 1 and DEMET PARLAK SÖNMEZ
More informationVolume 18 Figure 1. Notation 1. Notation 2. Observation 1. Remark 1. Remark 2. Remark 3. Remark 4. Remark 5. Remark 6. Theorem A [2]. Theorem B [2].
Bulletn of Mathematcal Scences and Applcatons Submtted: 016-04-07 ISSN: 78-9634, Vol. 18, pp 1-10 Revsed: 016-09-08 do:10.1805/www.scpress.com/bmsa.18.1 Accepted: 016-10-13 017 ScPress Ltd., Swtzerland
More informationREDUCTION MODULO p. We will prove the reduction modulo p theorem in the general form as given by exercise 4.12, p. 143, of [1].
REDUCTION MODULO p. IAN KIMING We wll prove the reducton modulo p theorem n the general form as gven by exercse 4.12, p. 143, of [1]. We consder an ellptc curve E defned over Q and gven by a Weerstraß
More informationk(k 1)(k 2)(p 2) 6(p d.
BLOCK-TRANSITIVE 3-DESIGNS WITH AFFINE AUTOMORPHISM GROUP Greg Gamble Let X = (Z p d where p s an odd prme and d N, and let B X, B = k. Then t was shown by Praeger that the set B = {B g g AGL d (p} s the
More informationCharacter Degrees of Extensions of PSL 2 (q) and SL 2 (q)
Character Degrees of Extensons of PSL (q) and SL (q) Donald L. Whte Department of Mathematcal Scences Kent State Unversty, Kent, Oho 444 E-mal: whte@math.kent.edu July 7, 01 Abstract Denote by S the projectve
More informationWeek 2. This week, we covered operations on sets and cardinality.
Week 2 Ths week, we covered operatons on sets and cardnalty. Defnton 0.1 (Correspondence). A correspondence between two sets A and B s a set S contaned n A B = {(a, b) a A, b B}. A correspondence from
More informationRandić Energy and Randić Estrada Index of a Graph
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL
More informationarxiv: v1 [math.co] 1 Mar 2014
Unon-ntersectng set systems Gyula O.H. Katona and Dánel T. Nagy March 4, 014 arxv:1403.0088v1 [math.co] 1 Mar 014 Abstract Three ntersecton theorems are proved. Frst, we determne the sze of the largest
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationSUCCESSIVE MINIMA AND LATTICE POINTS (AFTER HENK, GILLET AND SOULÉ) M(B) := # ( B Z N)
SUCCESSIVE MINIMA AND LATTICE POINTS (AFTER HENK, GILLET AND SOULÉ) S.BOUCKSOM Abstract. The goal of ths note s to present a remarably smple proof, due to Hen, of a result prevously obtaned by Gllet-Soulé,
More informationOdd automorphisms in vertex-transitive graphs
Also avalable at http://amc-journal.eu ISSN 1855-3966 (prnted edn.), ISSN 1855-3974 (electronc edn.) ARS MATHEMATICA CONTEMPORANEA 10 (2016) 427 437 Odd automorphsms n vertex-transtve graphs Ademr Hujdurovć,
More informationThe L(2, 1)-Labeling on -Product of Graphs
Annals of Pure and Appled Mathematcs Vol 0, No, 05, 9-39 ISSN: 79-087X (P, 79-0888(onlne Publshed on 7 Aprl 05 wwwresearchmathscorg Annals of The L(, -Labelng on -Product of Graphs P Pradhan and Kamesh
More informationComplete subgraphs in multipartite graphs
Complete subgraphs n multpartte graphs FLORIAN PFENDER Unverstät Rostock, Insttut für Mathematk D-18057 Rostock, Germany Floran.Pfender@un-rostock.de Abstract Turán s Theorem states that every graph G
More informationSolutions for Tutorial 1
Toc 1: Sem-drect roducts Solutons for Tutoral 1 1. Show that the tetrahedral grou s somorhc to the sem-drect roduct of the Klen four grou and a cyclc grou of order three: T = K 4 (Z/3Z). 2. Show further
More information12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product
12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA Here s an outlne of what I dd: (1) categorcal defnton (2) constructon (3) lst of basc propertes (4) dstrbutve property (5) rght exactness (6) localzaton
More informationOn the partial orthogonality of faithful characters. Gregory M. Constantine 1,2
On the partal orthogonalty of fathful characters by Gregory M. Constantne 1,2 ABSTRACT For conjugacy classes C and D we obtan an expresson for χ(c) χ(d), where the sum extends only over the fathful rreducble
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationDiscrete Mathematics
Dscrete Mathematcs 30 (00) 48 488 Contents lsts avalable at ScenceDrect Dscrete Mathematcs journal homepage: www.elsever.com/locate/dsc The number of C 3 -free vertces on 3-partte tournaments Ana Paulna
More informationALGEBRA HW 7 CLAY SHONKWILER
ALGEBRA HW 7 CLAY SHONKWILER 1 Whch of the followng rngs R are dscrete valuaton rngs? For those that are, fnd the fracton feld K = frac R, the resdue feld k = R/m (where m) s the maxmal deal), and a unformzer
More informationDiscrete Mathematics
Dscrete Mathematcs 32 (202) 720 728 Contents lsts avalable at ScVerse ScenceDrect Dscrete Mathematcs journal homepage: www.elsever.com/locate/dsc On the symmetrc dgraphs from powers modulo n Guxn Deng
More informationDiscrete Mathematics. Laplacian spectral characterization of some graphs obtained by product operation
Dscrete Mathematcs 31 (01) 1591 1595 Contents lsts avalable at ScVerse ScenceDrect Dscrete Mathematcs journal homepage: www.elsever.com/locate/dsc Laplacan spectral characterzaton of some graphs obtaned
More informationEdge Isoperimetric Inequalities
November 7, 2005 Ross M. Rchardson Edge Isopermetrc Inequaltes 1 Four Questons Recall that n the last lecture we looked at the problem of sopermetrc nequaltes n the hypercube, Q n. Our noton of boundary
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More information5 The Rational Canonical Form
5 The Ratonal Canoncal Form Here p s a monc rreducble factor of the mnmum polynomal m T and s not necessarly of degree one Let F p denote the feld constructed earler n the course, consstng of all matrces
More informationOn the smoothness and the totally strong properties for nearness frames
Int. Sc. Technol. J. Namba Vol 1, Issue 1, 2013 On the smoothness and the totally strong propertes for nearness frames Martn. M. Mugoch Department of Mathematcs, Unversty of Namba 340 Mandume Ndemufayo
More informationEvery planar graph is 4-colourable a proof without computer
Peter Dörre Department of Informatcs and Natural Scences Fachhochschule Südwestfalen (Unversty of Appled Scences) Frauenstuhlweg 31, D-58644 Iserlohn, Germany Emal: doerre(at)fh-swf.de Mathematcs Subject
More informationINVARIANT STABLY COMPLEX STRUCTURES ON TOPOLOGICAL TORIC MANIFOLDS
INVARIANT STABLY COMPLEX STRUCTURES ON TOPOLOGICAL TORIC MANIFOLDS HIROAKI ISHIDA Abstract We show that any (C ) n -nvarant stably complex structure on a topologcal torc manfold of dmenson 2n s ntegrable
More informationSpectral Graph Theory and its Applications September 16, Lecture 5
Spectral Graph Theory and ts Applcatons September 16, 2004 Lecturer: Danel A. Spelman Lecture 5 5.1 Introducton In ths lecture, we wll prove the followng theorem: Theorem 5.1.1. Let G be a planar graph
More informationNOTES FOR QUANTUM GROUPS, CRYSTAL BASES AND REALIZATION OF ŝl(n)-modules
NOTES FOR QUANTUM GROUPS, CRYSTAL BASES AND REALIZATION OF ŝl(n)-modules EVAN WILSON Quantum groups Consder the Le algebra sl(n), whch s the Le algebra over C of n n trace matrces together wth the commutator
More informationPolynomials. 1 More properties of polynomials
Polynomals 1 More propertes of polynomals Recall that, for R a commutatve rng wth unty (as wth all rngs n ths course unless otherwse noted), we defne R[x] to be the set of expressons n =0 a x, where a
More informationPAijpam.eu SOME NEW SUM PERFECT SQUARE GRAPHS S.G. Sonchhatra 1, G.V. Ghodasara 2
Internatonal Journal of Pure and Appled Mathematcs Volume 113 No. 3 2017, 489-499 ISSN: 1311-8080 (prnted verson); ISSN: 1314-3395 (on-lne verson) url: http://www.jpam.eu do: 10.12732/jpam.v1133.11 PAjpam.eu
More informationOn C 0 multi-contractions having a regular dilation
SUDIA MAHEMAICA 170 (3) (2005) On C 0 mult-contractons havng a regular dlaton by Dan Popovc (mşoara) Abstract. Commutng mult-contractons of class C 0 and havng a regular sometrc dlaton are studed. We prove
More informationAn Exposition on The Laws of Finite Pointed Groups
An Exposton on The Laws of Fnte Ponted Groups Justn Laverdure August 21, 2017 We outlne the detals of The Laws of Fnte Ponted Groups [1], wheren a fnte ponted group s constructed wth no fnte bass for ts
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationRepresentation theory and quantum mechanics tutorial Representation theory and quantum conservation laws
Representaton theory and quantum mechancs tutoral Representaton theory and quantum conservaton laws Justn Campbell August 1, 2017 1 Generaltes on representaton theory 1.1 Let G GL m (R) be a real algebrac
More informationA Note on \Modules, Comodules, and Cotensor Products over Frobenius Algebras"
Chn. Ann. Math. 27B(4), 2006, 419{424 DOI: 10.1007/s11401-005-0025-z Chnese Annals of Mathematcs, Seres B c The Edtoral Oce of CAM and Sprnger-Verlag Berln Hedelberg 2006 A Note on \Modules, Comodules,
More informationOn the Nilpotent Length of Polycyclic Groups
JOURNAL OF ALGEBRA 203, 125133 1998 ARTICLE NO. JA977321 On the Nlpotent Length of Polycyclc Groups Gerard Endmon* C.M.I., Unerste de Proence, UMR-CNRS 6632, 39, rue F. Jolot-Cure, 13453 Marselle Cedex
More informationEXPANSIVE MAPPINGS. by W. R. Utz
Volume 3, 978 Pages 6 http://topology.auburn.edu/tp/ EXPANSIVE MAPPINGS by W. R. Utz Topology Proceedngs Web: http://topology.auburn.edu/tp/ Mal: Topology Proceedngs Department of Mathematcs & Statstcs
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More informationCCO Commun. Comb. Optim.
Communcatons n Combnatorcs and Optmzaton Vol. 2 No. 2, 2017 pp.87-98 DOI: 10.22049/CCO.2017.13630 CCO Commun. Comb. Optm. Reformulated F-ndex of graph operatons Hamdeh Aram 1 and Nasrn Dehgard 2 1 Department
More informationKuroda s class number relation
ACTA ARITMETICA XXXV (1979) Kurodas class number relaton by C. D. WALTER (Dubln) Kurodas class number relaton [5] may be derved easly from that of Brauer [2] by elmnatng a certan module of unts, but the
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationA CHARACTERISATION OF VIRTUALLY FREE GROUPS
A CHARACTERISATION OF VIRTUALLY FREE GROUPS ROBERT H. GILMAN, SUSAN HERMILLER, DEREK F. HOLT, AND SARAH REES Abstract. We prove that a fntely generated group G s vrtually free f and only f there exsts
More informationThe Pseudoblocks of Endomorphism Algebras
Internatonal Mathematcal Forum, 4, 009, no. 48, 363-368 The Pseudoblocks of Endomorphsm Algebras Ahmed A. Khammash Department of Mathematcal Scences, Umm Al-Qura Unversty P.O.Box 796, Makkah, Saud Araba
More informationSmarandache-Zero Divisors in Group Rings
Smarandache-Zero Dvsors n Group Rngs W.B. Vasantha and Moon K. Chetry Department of Mathematcs I.I.T Madras, Chenna The study of zero-dvsors n group rngs had become nterestng problem snce 1940 wth the
More informationMATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS
MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples
More information2 More examples with details
Physcs 129b Lecture 3 Caltech, 01/15/19 2 More examples wth detals 2.3 The permutaton group n = 4 S 4 contans 4! = 24 elements. One s the dentty e. Sx of them are exchange of two objects (, j) ( to j and
More informationn ). This is tight for all admissible values of t, k and n. k t + + n t
MAXIMIZING THE NUMBER OF NONNEGATIVE SUBSETS NOGA ALON, HAROUT AYDINIAN, AND HAO HUANG Abstract. Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what
More informationGenericity of Critical Types
Genercty of Crtcal Types Y-Chun Chen Alfredo D Tllo Eduardo Fangold Syang Xong September 2008 Abstract Ely and Pesk 2008 offers an nsghtful characterzaton of crtcal types: a type s crtcal f and only f
More informationPerron Vectors of an Irreducible Nonnegative Interval Matrix
Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of
More information28 Finitely Generated Abelian Groups
8 Fntely Generated Abelan Groups In ths last paragraph of Chapter, we determne the structure of fntely generated abelan groups A complete classfcaton of such groups s gven Complete classfcaton theorems
More informationUNIQUE FACTORIZATION OF COMPOSITIVE HEREDITARY GRAPH PROPERTIES
UNIQUE FACTORIZATION OF COMPOSITIVE HEREDITARY GRAPH PROPERTIES IZAK BROERE AND EWA DRGAS-BURCHARDT Abstract. A graph property s any class of graphs that s closed under somorphsms. A graph property P s
More informationAli Omer Alattass Department of Mathematics, Faculty of Science, Hadramout University of science and Technology, P. O. Box 50663, Mukalla, Yemen
Journal of athematcs and Statstcs 7 (): 4448, 0 ISSN 5493644 00 Scence Publcatons odules n σ[] wth Chan Condtons on Small Submodules Al Omer Alattass Department of athematcs, Faculty of Scence, Hadramout
More informationA new construction of 3-separable matrices via an improved decoding of Macula s construction
Dscrete Optmzaton 5 008 700 704 Contents lsts avalable at ScenceDrect Dscrete Optmzaton journal homepage: wwwelsevercom/locate/dsopt A new constructon of 3-separable matrces va an mproved decodng of Macula
More informationMATH 5707 HOMEWORK 4 SOLUTIONS 2. 2 i 2p i E(X i ) + E(Xi 2 ) ä i=1. i=1
MATH 5707 HOMEWORK 4 SOLUTIONS CİHAN BAHRAN 1. Let v 1,..., v n R m, all lengths v are not larger than 1. Let p 1,..., p n [0, 1] be arbtrary and set w = p 1 v 1 + + p n v n. Then there exst ε 1,..., ε
More informationSemilattices of Rectangular Bands and Groups of Order Two.
1 Semlattces of Rectangular Bs Groups of Order Two R A R Monzo Abstract We prove that a semgroup S s a semlattce of rectangular bs groups of order two f only f t satsfes the dentty y y, y y, y S 1 Introducton
More informationSTATISTICAL GROUP THEORY
STATISTICAL GROUP THEORY ELAN BECHOR Abstract. Ths paper examnes two major results concernng the symmetrc group, S n. The frst result, Landau s theorem, gves an asymptotc formula for the maxmum order of
More informationTANGENT DIRAC STRUCTURES OF HIGHER ORDER. P. M. Kouotchop Wamba, A. Ntyam, and J. Wouafo Kamga
ARCHIVUM MATHEMATICUM BRNO) Tomus 47 2011), 17 22 TANGENT DIRAC STRUCTURES OF HIGHER ORDER P. M. Kouotchop Wamba, A. Ntyam, and J. Wouafo Kamga Abstract. Let L be an almost Drac structure on a manfold
More informationwhere a is any ideal of R. Lemma 5.4. Let R be a ring. Then X = Spec R is a topological space Moreover the open sets
5. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of
More informationLECTURE V. 1. More on the Chinese Remainder Theorem We begin by recalling this theorem, proven in the preceeding lecture.
LECTURE V EDWIN SPARK 1. More on the Chnese Remander Theorem We begn by recallng ths theorem, proven n the preceedng lecture. Theorem 1.1 (Chnese Remander Theorem). Let R be a rng wth deals I 1, I 2,...,
More informationTHE WEIGHTED WEAK TYPE INEQUALITY FOR THE STRONG MAXIMAL FUNCTION
THE WEIGHTED WEAK TYPE INEQUALITY FO THE STONG MAXIMAL FUNCTION THEMIS MITSIS Abstract. We prove the natural Fefferman-Sten weak type nequalty for the strong maxmal functon n the plane, under the assumpton
More informationMath 101 Fall 2013 Homework #7 Due Friday, November 15, 2013
Math 101 Fall 2013 Homework #7 Due Frday, November 15, 2013 1. Let R be a untal subrng of E. Show that E R R s somorphc to E. ANS: The map (s,r) sr s a R-balanced map of E R to E. Hence there s a group
More informationTHE CHVÁTAL-ERDŐS CONDITION AND 2-FACTORS WITH A SPECIFIED NUMBER OF COMPONENTS
Dscussones Mathematcae Graph Theory 27 (2007) 401 407 THE CHVÁTAL-ERDŐS CONDITION AND 2-FACTORS WITH A SPECIFIED NUMBER OF COMPONENTS Guantao Chen Department of Mathematcs and Statstcs Georga State Unversty,
More information20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The first idea is connectedness.
20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The frst dea s connectedness. Essentally, we want to say that a space cannot be decomposed
More informationCounterexamples to the Connectivity Conjecture of the Mixed Cells
Dscrete Comput Geom 2:55 52 998 Dscrete & Computatonal Geometry 998 Sprnger-Verlag New York Inc. Counterexamples to the Connectvty Conjecture of the Mxed Cells T. Y. L and X. Wang 2 Department of Mathematcs
More informationAnti-van der Waerden numbers of 3-term arithmetic progressions.
Ant-van der Waerden numbers of 3-term arthmetc progressons. Zhanar Berkkyzy, Alex Schulte, and Mchael Young Aprl 24, 2016 Abstract The ant-van der Waerden number, denoted by aw([n], k), s the smallest
More informationDIFFERENTIAL SCHEMES
DIFFERENTIAL SCHEMES RAYMOND T. HOOBLER Dedcated to the memory o Jerry Kovacc 1. schemes All rngs contan Q and are commutatve. We x a d erental rng A throughout ths secton. 1.1. The topologcal space. Let
More information( 1) i [ d i ]. The claim is that this defines a chain complex. The signs have been inserted into the definition to make this work out.
Mon, Apr. 2 We wsh to specfy a homomorphsm @ n : C n ()! C n (). Snce C n () s a free abelan group, the homomorphsm @ n s completely specfed by ts value on each generator, namely each n-smplex. There are
More informationThe Second Eigenvalue of Planar Graphs
Spectral Graph Theory Lecture 20 The Second Egenvalue of Planar Graphs Danel A. Spelman November 11, 2015 Dsclamer These notes are not necessarly an accurate representaton of what happened n class. The
More informationREGULAR POSITIVE TERNARY QUADRATIC FORMS. 1. Introduction
REGULAR POSITIVE TERNARY QUADRATIC FORMS BYEONG-KWEON OH Abstract. A postve defnte quadratc form f s sad to be regular f t globally represents all ntegers that are represented by the genus of f. In 997
More informationJournal of Algebra 368 (2012) Contents lists available at SciVerse ScienceDirect. Journal of Algebra.
Journal of Algebra 368 (2012) 70 74 Contents lsts avalable at ScVerse ScenceDrect Journal of Algebra www.elsever.com/locate/jalgebra An algebro-geometrc realzaton of equvarant cohomology of some Sprnger
More informationExpected Value and Variance
MATH 38 Expected Value and Varance Dr. Neal, WKU We now shall dscuss how to fnd the average and standard devaton of a random varable X. Expected Value Defnton. The expected value (or average value, or
More information9 Characteristic classes
THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct
More informationYong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 )
Kangweon-Kyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROW-ACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often
More informationMTH 819 Algebra I S13. Homework 1/ Solutions. 1 if p n b and p n+1 b 0 otherwise ) = 0 if p q or n m. W i = rw i
MTH 819 Algebra I S13 Homework 1/ Solutons Defnton A. Let R be PID and V a untary R-module. Let p be a prme n R and n Z +. Then d p,n (V) = dm R/Rp p n 1 Ann V (p n )/p n Ann V (p n+1 ) Note here that
More informationLecture Notes Introduction to Cluster Algebra
Lecture Notes Introducton to Cluster Algebra Ivan C.H. Ip Updated: Ma 7, 2017 3 Defnton and Examples of Cluster algebra 3.1 Quvers We frst revst the noton of a quver. Defnton 3.1. A quver s a fnte orented
More informationTHE ORNSTEIN-WEISS LEMMA FOR DISCRETE AMENABLE GROUPS.
THE ORNSTEIN-WEISS LEMMA FOR DISCRETE AMENABLE GROUPS FABRICE KRIEGER Abstract In ths note we prove a convergence theorem for nvarant subaddtve functons defned on the fnte subsets of a dscrete amenable
More informationU.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016
U.C. Berkeley CS94: Spectral Methods and Expanders Handout 8 Luca Trevsan February 7, 06 Lecture 8: Spectral Algorthms Wrap-up In whch we talk about even more generalzatons of Cheeger s nequaltes, and
More informationExercise Solutions to Real Analysis
xercse Solutons to Real Analyss Note: References refer to H. L. Royden, Real Analyss xersze 1. Gven any set A any ɛ > 0, there s an open set O such that A O m O m A + ɛ. Soluton 1. If m A =, then there
More informationA New Refinement of Jacobi Method for Solution of Linear System Equations AX=b
Int J Contemp Math Scences, Vol 3, 28, no 17, 819-827 A New Refnement of Jacob Method for Soluton of Lnear System Equatons AX=b F Naem Dafchah Department of Mathematcs, Faculty of Scences Unversty of Gulan,
More informationPiecewise Linear Parametrization of Canonical Bases
Pecewse Lnear Parametrzaton of Canoncal Bases The MIT Faculty has made ths artcle openly avalle. Please share how ths access benefts you. Your story matters. Ctaton As Publshed Publsher Lusztg, G. Pecewse
More informationA combinatorial proof of multiple angle formulas involving Fibonacci and Lucas numbers
Notes on Number Theory and Dscrete Mathematcs ISSN 1310 5132 Vol. 20, 2014, No. 5, 35 39 A combnatoral proof of multple angle formulas nvolvng Fbonacc and Lucas numbers Fernando Córes 1 and Dego Marques
More informationarxiv: v1 [math.co] 7 Apr 2015
Ranbow connecton n some dgraphs Jesús Alva-Samos 1 Juan José Montellano-Ballesteros Abstract arxv:1504.0171v1 [math.co] 7 Apr 015 An edge-coloured graph G s ranbow connected f any two vertces are connected
More informationDeriving the X-Z Identity from Auxiliary Space Method
Dervng the X-Z Identty from Auxlary Space Method Long Chen Department of Mathematcs, Unversty of Calforna at Irvne, Irvne, CA 92697 chenlong@math.uc.edu 1 Iteratve Methods In ths paper we dscuss teratve
More information