A Class of Z4C-Groups
|
|
- Adrian Robert Wade
- 5 years ago
- Views:
Transcription
1 Applied Mathematical Sciences, Vol. 9, 2015, no. 41, HIKARI Ltd, A Class of Z4C-Groups Jinshan Zhang 1 School of Science Sichuan University of Science and Engineering Zigong, , P. R. China Dandan Liu School of Mechanical Engineering Science Sichuan University of Science and Engineering Zigong, , P. R. China Copyright c 2014 Jinshan Zhang and Dandan Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The aim of this note is to classify a class of finite solvable groups whose every irreducible character vanish on at most four conjugacy classes in the character table. 1 Introduction Let G be a finite group and υ(χ) := {g G χ(g) = 0}, where χ is an irreducible complex character of G. A classical theorem of Burnside asserts that υ(χ) is non-empty for all χ Irr 1 (G), where Irr 1 (G) denotes the set of non-linear irreducible complex characters of G. We say that an element g G is a vanishing element of G if there exists χ in Irr(G) such that χ(g) = 0, and otherwise we call x a non-vanishing element. Let Van(G) denote the set of vanishing elements of G, that is, 1 Corresponding author Van(G) = {g G χ(g) = 0 for some χ Irr(G)}.
2 2032 Jinshan Zhang and Dandan Liu Clearly, Van(G) is a proper normal subset of G. We denote k G (N) the number of conjugacy classes of G contained in N, where N is a normal subset of G. Recently, in [4] we classify the finite groups whose every irreducible character of G vanishes on at most three conjugacy classes. Furthermore, the finite groups with k G (Van(G)) 4 have been studied (see [5]). Clearly, if G is a V nc-group, then in particular every irreducible character of G vanishes on at most n conjugacy classes of G. In this paper, we classify a class of finite solvable groups whose every irreducible character vanish on at most four conjugacy classes in the character table. Generally, we define Definition 1.1. A group G is called a ZnC-group if every irreducible character vanishes on at most n conjugacy classes of G. Remark 1.2. A V 4C-group is a Z4C-group. but the converse is not true. For example, G = a, b a 8 = b 2 = 1, b 1 ab = a 1. The Z3C-groups have been very well characterized (see [4]). result of this paper are as follows. The main Theorem A. Let N be a subgroup of G such that G : N = 2. Assume that G is non-nilpotent and that N is abelian. Then G is a Z4C-group but not a Z3C-group if and only if G = G P, where G is a normal abelian 2-complement of G, P Syl 2 (G), P = 8, Z(G) = 4, and G/Z(G) is a Frobenius group with kernel (G/Z(G)) = G and complement P/Z(G) of order 2. In this paper, G always denotes a finite group. Notation is standard and taken from [1]. In particular, denote k G (N) the number of conjugacy classes of G contained in N, where N is a normal subset of G. For N G, set Irr(G N) = Irr(G)\Irr(G/N). 2 Proof of Theorem A We will use frequently the following lemma (see [3, Theorem 2.1]). Lemma 2.1. Let G be non-abelian, and let χ Irr 1 (G). Assume that N is a normal subgroup of G such that G N < G. If χ N is not irreducible, then the following two statements hold: (1) There exists a normal subgroup H of G such that N H < G and G\H υ(χ). (2) If (G\G ) υ(χ) consists of n conjugacy classes of G, then [H : G ] ([G : H] 1) n.
3 A class of Z4C-groups 2033 The following Lemma characterizes the group G when its normal subgroup N contains all but two conjugacy classes. We only need and record the following weaker description of this classification. Lemma 2.2. ([2, Theorem 2.2]). Let N be a normal subgroup of a nonabelian solvable group G. Then k G (G\N) = 2 if and only if G is one of the following solvable groups. (1) N = 1 and G = S 3. (2) G/N = 3 and G is a Frobenius group with kernel N. (3) G/N = 2 and C G (x) = 4 for all x G N. Proposition 2.3. ([5, Theorem 2.3]). Suppose that G is a non-abelian nilpotent group. If G is a Z4C-group, then G is one of the following groups: (1) G = D 8 or Q 8. (2) G = a, b a 8 = b 2 = 1, b 1 ab = a 1. (3) G = a, b a 8 = 1, b 2 = a 4, b 1 ab = a 1. (4) G = a, b a 8 = b 2 = 1, b 1 ab = a 3. Lemma 2.4. ([4, Lemma 2.7]). Let G be a meta-abelian group. If [G : G ] = p, then G is a Frobenius group with kernel G and complement of order p. Proof of Theorem A. Since G : N = 2 and N is abelian, G = KP, where K is a normal abelian 2-complement of G and P Syl 2 (G). If G/K = P is non-abelian, then by Proposition 2.3, G/K satisfies types (1), (2), (3) or (4) in Proposition 2.3. Set N/K = (G/K). Then [G : N] = 4. Take an irreducible character ξ of G/K with υ(ξ) = G\N. Then it follows by Lemma 2.1 that N = G. Suppose that G := G/K is of order 16. Then k G (υ(ξ)) = 4, and thus the hypothesis yields that k G (υ(ξ)) = k G (υ(ξ)) = 4, and so χ vanishes only on υ(ξ) for every χ Irr(G K). Observe that C G (g) = 4 or 8 for every g G\G. Then we easily obtain a contradiction (note that [N : G ] = 2 and N is abelian). Suppose that G := G/K is of order 8. Then k G (υ(ξ)) = 3. If k G (υ(ξ)) = 3, then arguing as the above paragraph, we also obtain a contradiction. Hence k G (υ(ξ)) = 4. Observe that there exists an element g in G\G such that C G (g) = 6. On the other hand, we easily see that Z(G) = 2, thus it is easy to conclude that 4 divides C G (z), a contradiction. Hence we may suppose that P is abelian. Note that P is abelian, consequently, G K. Applying Lemma 2.1, we conclude that P 8 and that one of the following two cases occurs: (i) F (G) : G = 1, 2 or 4; (ii) F (G) : G = 3.
4 2034 Jinshan Zhang and Dandan Liu Case 1. N : G = 3. Recall now that G K N, then K = N and so P = 2. Observe that G\N = xg + yg + zg, where x, y, z G\N. Suppose that k G (G\N) = 3. We have G\N = G C G (x) + G C G (y) + G C G (z), and C G (x) = C G (y) = 6 = C G (y) = 6. On the other hand, by the second orthogonality relation we have 6 = C G (g) = G/G + { χ(g) 2 χ Irr 1 (G)}, for all g G\N. Hence χ(g) = 0 for all g G\N and all χ Irr 1 (G). Let P =< t >, where t is an involution. By Fitting Lemma, we have N = C N (P ) [N, P ]. Obviously, C N) (t) = C N (P ) = Z(G). Since C G (g) = 6 for every g G\N, we conclude that Z(G) = 3. So, G = B Z(G), where B = [N, P ]P. Observe that B is a Frobenius group with kernel B = G = [N, P ] and complement of order 2. Then G is a Z3C-group. If k G (G\N) = 4, we easily see that there exists an element g in G\G such that 4 divides C G (z), a contradiction. Case 2. N : G = 1, 2 or 4. First, assume that N = G. Since G is abelian, it follows by Lemma 3.2 that G is a Frobenius group with kernel G and complement of order 2. Thus G is a Z3C-group. Second, assume that N : G = 2. Then G : G = 4, and thus G = G P, where G is a normal 2-complement of G and P = 4. Observe that G/O 2 (G) is a Frobenius group with Frobenius kernel (G/O 2 (G)) = G and complement of order 2. Clearly, G is a Z3C-group. Finally, assume that N : G = 4. Thus G : G = 8 and so G = G P, where G is a normal 2-complement of G and P = 8. Clearly, O 2 (G) = 4, and thus G/O 2 (G) : (G/O 2 (G)) = 2. Furthermore, applying again Lemma 2.4, we see that G/O 2 (G) is a Frobenius group with Frobenius kernel (G/O 2 (G)) = G and complement of order 2, and we are done. Acknowledgment. This paper is supported by the NNSF of China ( , ), the Opening Project of Sichuan Province University Key Laborary Bridge Non-destruction Detecting and Engineering Computing (2013QYJ02) and the Sichuan Provincial Education Department Foundation of China (12ZB291).
5 A class of Z4C-groups 2035 Reference [1] I. M. Isaacs. Character theory of finite groups (Academic Prees, 1976). [2] G. Qian, W. Shi and X. You. Conjugacy classes outside a normal subgroup. Comm. Algebra 12 (2004), [3] Y. C. Ren, X. H. Liu and J. S. Zhang. On restriction and zeros of characters of finite groups. J. Sichuan Univ. (Natural Science) 44 (2007), [4] J. S. Zhang, J. T. Shi and Z. C. Shen. Finite groups in which every irreducible character vanishes on at most three conjugacy classes. J. Group Theory 13 (2010), [5] J. S. Zhang, Z. C. Shen and J. T. Shi. Finite groups with few vanishing elements. to appear in Glasnik Matematicki. Received: December 17, 2014; Pubslihed: March 14, 2015
A New Characterization of A 11
International Journal of Algebra, Vol. 8, 2014, no. 6, 253-266 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2014.4211 A New Characterization of A 11 Yong Yang, Shitian Liu and Yanhua Huang
More informationFinite Groups with ss-embedded Subgroups
International Journal of Algebra, Vol. 11, 2017, no. 2, 93-101 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2017.7311 Finite Groups with ss-embedded Subgroups Xinjian Zhang School of Mathematical
More informationOn Regular Prime Graphs of Solvable Groups
International Journal of Algebra, Vol. 10, 2016, no. 10, 491-495 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2016.6858 On Regular Prime Graphs of Solvable Groups Donnie Munyao Kasyoki Department
More informationUnit Group of Z 2 D 10
International Journal of Algebra, Vol. 9, 2015, no. 4, 179-183 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2015.5420 Unit Group of Z 2 D 10 Parvesh Kumari Department of Mathematics Indian
More informationCLASSIFYING CAMINA GROUPS: ATHEOREMOFDARKANDSCOPPOLA
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 44, Number 2, 2014 CLASSIFYING CAMINA GROUPS: ATHEOREMOFDARKANDSCOPPOLA MARK L. LEWIS ABSTRACT. Recall that a group G is a Camina group if every nonlinear irreducible
More informationClassifying Camina groups: A theorem of Dark and Scoppola
Classifying Camina groups: A theorem of Dark and Scoppola arxiv:0807.0167v5 [math.gr] 28 Sep 2011 Mark L. Lewis Department of Mathematical Sciences, Kent State University Kent, Ohio 44242 E-mail: lewis@math.kent.edu
More informationA WEAKER QUANTITATIVE CHARACTERIZATION OF THE SPORADIC SIMPLE GROUPS
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 39 2018 (45 54) 45 A WEAKER QUANTITATIVE CHARACTERIZATION OF THE SPORADIC SIMPLE GROUPS Jinbao Li Department of Mathematics Chongqing University of Arts
More informationInternational Journal of Pure and Applied Mathematics Volume 13 No , M-GROUP AND SEMI-DIRECT PRODUCT
International Journal of Pure and Applied Mathematics Volume 13 No. 3 2004, 381-389 M-GROUP AND SEMI-DIRECT PRODUCT Liguo He Department of Mathematics Shenyang University of Technology Shenyang, 110023,
More informationONZEROSOFCHARACTERSOFFINITEGROUPS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 4, April 1999, Pages 977 983 S 0002-9939(99)04790-5 ONZEROSOFCHARACTERSOFFINITEGROUPS DAVID CHILLAG (Communicated by Ronald M. Solomon)
More informationSome Properties of D-sets of a Group 1
International Mathematical Forum, Vol. 9, 2014, no. 21, 1035-1040 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.45104 Some Properties of D-sets of a Group 1 Joris N. Buloron, Cristopher
More informationDirect Product of BF-Algebras
International Journal of Algebra, Vol. 10, 2016, no. 3, 125-132 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.614 Direct Product of BF-Algebras Randy C. Teves and Joemar C. Endam Department
More informationJoin Reductions and Join Saturation Reductions of Abstract Knowledge Bases 1
International Journal of Contemporary Mathematical Sciences Vol. 12, 2017, no. 3, 109-115 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2017.7312 Join Reductions and Join Saturation Reductions
More informationLandau s Theorem for π-blocks of π-separable groups
Landau s Theorem for π-blocks of π-separable groups Benjamin Sambale October 13, 2018 Abstract Slattery has generalized Brauer s theory of p-blocks of finite groups to π-blocks of π-separable groups where
More informationBlock-Transitive 4 (v, k, 4) Designs and Suzuki Groups
International Journal of Algebra, Vol. 10, 2016, no. 1, 27-32 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.51277 Block-Transitive 4 (v, k, 4) Designs and Suzuki Groups Shaojun Dai Department
More informationOn Symmetric Bi-Multipliers of Lattice Implication Algebras
International Mathematical Forum, Vol. 13, 2018, no. 7, 343-350 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2018.8423 On Symmetric Bi-Multipliers of Lattice Implication Algebras Kyung Ho
More informationThe C 8 -Group Having Five Maximal Subgroups of Index 2 and Three of Index 3
International Journal of Algebra, Vol. 11, 2017, no. 8, 375-379 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2017.71047 The C 8 -Group Having Five Maximal Subgroups of Index 2 and Three of
More informationOn Annihilator Small Intersection Graph
International Journal of Contemporary Mathematical Sciences Vol. 12, 2017, no. 7, 283-289 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2017.7931 On Annihilator Small Intersection Graph Mehdi
More informationA PROOF OF BURNSIDE S p a q b THEOREM
A PROOF OF BURNSIDE S p a q b THEOREM OBOB Abstract. We prove that if p and q are prime, then any group of order p a q b is solvable. Throughout this note, denote by A the set of algebraic numbers. We
More informationCHARACTER DEGREE SUMS IN FINITE NONSOLVABLE GROUPS
CHARACTER DEGREE SUMS IN FINITE NONSOLVABLE GROUPS KAY MAGAARD AND HUNG P. TONG-VIET Abstract. Let N be a minimal normal nonabelian subgroup of a finite group G. We will show that there exists a nontrivial
More informationOn the solvability of groups with four class sizes.
On the solvability of groups with four class sizes. Antonio Beltrán Departamento de Matemáticas, Universidad Jaume I, 12071 Castellón, Spain e-mail: abeltran@mat.uji.es and María José Felipe Instituto
More informationPOS-GROUPS WITH SOME CYCLIC SYLOW SUBGROUPS. Communicated by Ali Reza Ashrafi. 1. Introduction
Bulletin of the Iranian Mathematical Society Vol. 39 No. 5 (2013), pp 941-957. POS-GROUPS WITH SOME CYCLIC SYLOW SUBGROUPS R. SHEN, W. SHI AND J. SHI Communicated by Ali Reza Ashrafi Abstract. A finite
More informationFinite groups with many values in a column or a row of the character table
Publ. Math. Debrecen 69/3 (006), 81 90 Finite groups with many values in a column or a row of the character table By MARIAGRAZIA BIANCHI (Milano), DAVID CHILLAG (Haifa) and ANNA GILLIO (Milano) This paper
More informationFinite groups determined by an inequality of the orders of their elements
Publ. Math. Debrecen 80/3-4 (2012), 457 463 DOI: 10.5486/PMD.2012.5168 Finite groups determined by an inequality of the orders of their elements By MARIUS TĂRNĂUCEANU (Iaşi) Abstract. In this note we introduce
More informationSolution of Brauer s k(b)-conjecture for π-blocks of π-separable groups
Solution of Brauer s k(b)-conjecture for π-blocks of π-separable groups Benjamin Sambale October 9, 2018 Abstract Answering a question of Pálfy and Pyber, we first prove the following extension of the
More informationA Note on Finite Groups in which C-Normality is a Transitive Relation
International Mathematical Forum, Vol. 8, 2013, no. 38, 1881-1887 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2013.39168 A Note on Finite Groups in which C-Normality is a Transitive Relation
More informationThe Rainbow Connection of Windmill and Corona Graph
Applied Mathematical Sciences, Vol. 8, 2014, no. 128, 6367-6372 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.48632 The Rainbow Connection of Windmill and Corona Graph Yixiao Liu Department
More informationCHARACTER THEORY OF FINITE GROUPS. Chapter 1: REPRESENTATIONS
CHARACTER THEORY OF FINITE GROUPS Chapter 1: REPRESENTATIONS G is a finite group and K is a field. A K-representation of G is a homomorphism X : G! GL(n, K), where GL(n, K) is the group of invertible n
More informationPrime Hyperideal in Multiplicative Ternary Hyperrings
International Journal of Algebra, Vol. 10, 2016, no. 5, 207-219 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.6320 Prime Hyperideal in Multiplicative Ternary Hyperrings Md. Salim Department
More informationH-Transversals in H-Groups
International Journal of Algebra, Vol. 8, 2014, no. 15, 705-712 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2014.4885 H-Transversals in H-roups Swapnil Srivastava Department of Mathematics
More informationA Generalization of p-rings
International Journal of Algebra, Vol. 9, 2015, no. 8, 395-401 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2015.5848 A Generalization of p-rings Adil Yaqub Department of Mathematics University
More informationMorphisms Between the Groups of Semi Magic Squares and Real Numbers
International Journal of Algebra, Vol. 8, 2014, no. 19, 903-907 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2014.212137 Morphisms Between the Groups of Semi Magic Squares and Real Numbers
More informationOn Geometric Hyper-Structures 1
International Mathematical Forum, Vol. 9, 2014, no. 14, 651-659 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.312232 On Geometric Hyper-Structures 1 Mashhour I.M. Al Ali Bani-Ata, Fethi
More informationApproximations to the t Distribution
Applied Mathematical Sciences, Vol. 9, 2015, no. 49, 2445-2449 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.52148 Approximations to the t Distribution Bashar Zogheib 1 and Ali Elsaheli
More informationThe Influence of Minimal Subgroups on the Structure of Finite Groups 1
Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 14, 675-683 The Influence of Minimal Subgroups on the Structure of Finite Groups 1 Honggao Zhang 1, Jianhong Huang 1,2 and Yufeng Liu 3 1. Department
More informationRainbow Connection Number of the Thorn Graph
Applied Mathematical Sciences, Vol. 8, 2014, no. 128, 6373-6377 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.48633 Rainbow Connection Number of the Thorn Graph Yixiao Liu Department
More informationDouble Total Domination in Circulant Graphs 1
Applied Mathematical Sciences, Vol. 12, 2018, no. 32, 1623-1633 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.811172 Double Total Domination in Circulant Graphs 1 Qin Zhang and Chengye
More informationInternational Journal of Algebra, Vol. 7, 2013, no. 3, HIKARI Ltd, On KUS-Algebras. and Areej T.
International Journal of Algebra, Vol. 7, 2013, no. 3, 131-144 HIKARI Ltd, www.m-hikari.com On KUS-Algebras Samy M. Mostafa a, Mokhtar A. Abdel Naby a, Fayza Abdel Halim b and Areej T. Hameed b a Department
More informationβ Baire Spaces and β Baire Property
International Journal of Contemporary Mathematical Sciences Vol. 11, 2016, no. 5, 211-216 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2016.612 β Baire Spaces and β Baire Property Tugba
More informationDouble Total Domination on Generalized Petersen Graphs 1
Applied Mathematical Sciences, Vol. 11, 2017, no. 19, 905-912 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.7114 Double Total Domination on Generalized Petersen Graphs 1 Chengye Zhao 2
More informationOn a Diophantine Equation 1
International Journal of Contemporary Mathematical Sciences Vol. 12, 2017, no. 2, 73-81 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2017.728 On a Diophantine Equation 1 Xin Zhang Department
More informationSylow 2-Subgroups of Solvable Q-Groups
E extracta mathematicae Vol. 22, Núm. 1, 83 91 (2007) Sylow 2-Subgroups of Solvable Q-roups M.R. Darafsheh, H. Sharifi Department of Mathematics, Statistics and Computer Science, Faculty of Science University
More informationOn the Probability that a Group Element Fixes a Set and its Generalized Conjugacy Class Graph
International Journal of Mathematical Analysis Vol. 9, 2015, no. 4, 161-167 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.411336 On the Probability that a roup Element Fixes a Set and
More informationNON-NILPOTENT GROUPS WITH THREE CONJUGACY CLASSES OF NON-NORMAL SUBGROUPS. Communicated by Alireza Abdollahi. 1. Introduction
International Journal of Group Theory ISSN (print): 2251-7650, ISSN (on-line): 2251-7669 Vol. 3 No. 2 (2014), pp. 1-7. c 2014 University of Isfahan www.theoryofgroups.ir www.ui.ac.ir NON-NILPOTENT GROUPS
More informationOn the nilpotent conjugacy class graph of groups
Note di Matematica ISSN 1123-2536, e-issn 1590-0932 Note Mat. 37 (2017) no. 2, 77 89. doi:10.1285/i15900932v37n2p77 On the nilpotent conjugacy class graph of groups A. Mohammadian Department of Pure Mathematics,
More informationCommunications in Algebra Publication details, including instructions for authors and subscription information:
This article was downloaded by: [Professor Alireza Abdollahi] On: 04 January 2013, At: 19:35 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered
More informationNAVARRO VERTICES AND NORMAL SUBGROUPS IN GROUPS OF ODD ORDER
NAVARRO VERTICES AND NORMAL SUBGROUPS IN GROUPS OF ODD ORDER JAMES P. COSSEY Abstract. Let p be a prime and suppose G is a finite solvable group and χ is an ordinary irreducible character of G. Navarro
More informationIrreducible characters taking root of unity values on p-singular elements
Irreducible characters taking root of unity values on p-singular elements by Gabriel Navarro Departament d Àlgebra Universitat de València 46100 Burjassot SPAIN E-mail: gabriel.navarro@uv.es and Geoffrey
More informationThe Automorphisms of a Lie algebra
Applied Mathematical Sciences Vol. 9 25 no. 3 2-27 HIKARI Ltd www.m-hikari.com http://dx.doi.org/.2988/ams.25.4895 The Automorphisms of a Lie algebra WonSok Yoo Department of Applied Mathematics Kumoh
More informationOn the Power of Standard Polynomial to M a,b (E)
International Journal of Algebra, Vol. 10, 2016, no. 4, 171-177 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.6214 On the Power of Standard Polynomial to M a,b (E) Fernanda G. de Paula
More informationLeft R-prime (R, S)-submodules
International Mathematical Forum, Vol. 8, 2013, no. 13, 619-626 HIKARI Ltd, www.m-hikari.com Left R-prime (R, S)-submodules T. Khumprapussorn Department of Mathematics, Faculty of Science King Mongkut
More informationRestrained Independent 2-Domination in the Join and Corona of Graphs
Applied Mathematical Sciences, Vol. 11, 2017, no. 64, 3171-3176 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.711343 Restrained Independent 2-Domination in the Join and Corona of Graphs
More informationA SURVEY ON THE ESTIMATION OF COMMUTATIVITY IN FINITE GROUPS
A SURVEY ON THE ESTIMATION OF COMMUTATIVITY IN FINITE GROUPS A K DAS, R K NATH, AND M R POURNAKI Abstract Let G be a finite group and let C = {(x, y G G xy = yx} Then Pr(G = C / G 2 is the probability
More informationSelected exercises from Abstract Algebra by Dummit and Foote (3rd edition).
Selected exercises from Abstract Algebra by Dummit Foote (3rd edition). Bryan Félix Abril 12, 2017 Section 4.1 Exercise 1. Let G act on the set A. Prove that if a, b A b = ga for some g G, then G b = gg
More informationMoore-Penrose Inverses of Operators in Hilbert C -Modules
International Journal of Mathematical Analysis Vol. 11, 2017, no. 8, 389-396 HIKARI Ltd, www.m-hikari.com https//doi.org/10.12988/ijma.2017.7342 Moore-Penrose Inverses of Operators in Hilbert C -Modules
More informationA dual version of Huppert s conjecture on conjugacy class sizes
A dual version of Huppert s conjecture on conjugacy class sizes Zeinab Akhlaghi 1, Maryam Khatami 2, Tung Le 3, Jamshid Moori 3, Hung P. Tong-Viet 4 1 Faculty of Math. and Computer Sci., Amirkabir University
More informationOrder-theoretical Characterizations of Countably Approximating Posets 1
Int. J. Contemp. Math. Sciences, Vol. 9, 2014, no. 9, 447-454 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2014.4658 Order-theoretical Characterizations of Countably Approximating Posets
More informationOn a Certain Representation in the Pairs of Normed Spaces
Applied Mathematical Sciences, Vol. 12, 2018, no. 3, 115-119 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.712362 On a Certain Representation in the Pairs of ormed Spaces Ahiro Hoshida
More informationCHARACTER EXPANSIVENESS IN FINITE GROUPS
CHARACTER EXPANSIVENESS IN FINITE GROUPS Abstract. We say that a finite group G is conjugacy expansive if for any normal subset S and any conjugacy class C of G the normal set SC consists of at least as
More informationUniversität Kassel, Universitat de València, University of Copenhagen
ZEROS OF CHARACTERS OF FINITE GROUPS Gunter Malle, Gabriel Navarro and Jørn B. Olsson Universität Kassel, Universitat de València, University of Copenhagen 1. Introduction Let G be a finite group and let
More informationMathematical Journal of Okayama University
Mathematical Journal of Okayama University Volume 48, Issue 1 2006 Article 8 JANUARY 2006 Characterization of Frobenius Groups of Special Type Arun S. Muktibodh Mohota Science College Copyright c 2006
More informationNonsolvable Groups with No Prime Dividing Three Character Degrees
Nonsolvable Groups with No Prime Dividing Three Character Degrees Mark L. Lewis and Donald L. White Department of Mathematical Sciences, Kent State University Kent, Ohio 44242 E-mail: lewis@math.kent.edu,
More informationAlgebra Exercises in group theory
Algebra 3 2010 Exercises in group theory February 2010 Exercise 1*: Discuss the Exercises in the sections 1.1-1.3 in Chapter I of the notes. Exercise 2: Show that an infinite group G has to contain a non-trivial
More informationRecognition of Some Symmetric Groups by the Set of the Order of Their Elements
Recognition of Some Symmetric Groups by the Set of the Order of Their Elements A. R. Moghaddamfar, M. R. Pournaki * Abstract For a finite group G, let π e (G) be the set of order of elements in G and denote
More informationSOLVABLE FUSION CATEGORIES AND A CATEGORICAL BURNSIDE S THEOREM
SOLVABLE FUSION CATEGORIES AND A CATEGORICAL BURNSIDE S THEOREM PAVEL ETINGOF The goal of this talk is to explain the classical representation-theoretic proof of Burnside s theorem in finite group theory,
More informationKevin James. p-groups, Nilpotent groups and Solvable groups
p-groups, Nilpotent groups and Solvable groups Definition A maximal subgroup of a group G is a proper subgroup M G such that there are no subgroups H with M < H < G. Definition A maximal subgroup of a
More informationThe Endomorphism Ring of a Galois Azumaya Extension
International Journal of Algebra, Vol. 7, 2013, no. 11, 527-532 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2013.29110 The Endomorphism Ring of a Galois Azumaya Extension Xiaolong Jiang
More informationCharacters and quasi-permutation representations
International Mathematical Forum, 1, 2006, no. 37, 1819-1824 Characters and quasi-permutation representations of finite p-groups with few non-normal subgroups Houshang Behravesh and Sebar Ghadery Department
More informationTRANSITIVE PERMUTATION GROUPS IN WHICH ALL DERANGEMENTS ARE INVOLUTIONS
TRANSITIVE PERMUTATION GROUPS IN WHICH ALL DERANGEMENTS ARE INVOLUTIONS I. M. Isaacs Department of Mathematics, University of Wisconsin Madison, WI 53706 USA e-mail: isaacs@math.wisc.edu Thomas Michael
More informationGroup Gradings on Finite Dimensional Lie Algebras
Algebra Colloquium 20 : 4 (2013) 573 578 Algebra Colloquium c 2013 AMSS CAS & SUZHOU UNIV Group Gradings on Finite Dimensional Lie Algebras Dušan Pagon Faculty of Natural Sciences and Mathematics, University
More informationOn Boolean Like Ring Extension of a Group
International Journal of Algebra, Vol. 8, 2014, no. 3, 121-128 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2014.312136 On Boolean Like Ring Extension of a Group Dawit Chernet and K. Venkateswarlu
More informationGeneralized Boolean and Boolean-Like Rings
International Journal of Algebra, Vol. 7, 2013, no. 9, 429-438 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2013.2894 Generalized Boolean and Boolean-Like Rings Hazar Abu Khuzam Department
More informationTransitivity of properties of two-generator subgroups of finite groups
Transitivity of properties of two-generator subgroups of finite groups Primož Moravec University of Ljubljana (joint work with Costantino Delizia and Chiara Nicotera) Monash University, 2016 (visit funded
More informationResearch Article Commutator Length of Finitely Generated Linear Groups
Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 008, Article ID 81734, 5 pages doi:10.1155/008/81734 Research Article Commutator Length of Finitely
More informationDiophantine Equations. Elementary Methods
International Mathematical Forum, Vol. 12, 2017, no. 9, 429-438 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.7223 Diophantine Equations. Elementary Methods Rafael Jakimczuk División Matemática,
More informationRecognizing the Automorphism Groups of Mathieu Groups Through Their Orders and Large Degrees of Their Irreducible Characters
Chin. Ann. Math. 37B(4), 2016, 495 502 DOI: 10.1007/s11401-016-1024-y Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2016 Recognizing the Automorphism
More informationON THE PRIME GRAPHS OF THE AUTOMORPHISM GROUPS OF SPORADIC SIMPLE GROUPS. Behrooz Khosravi
ARCHIVUM MATHEMATICUM (BRNO) Tomus 45 (2009), 83 94 ON THE PRIME GRAPHS OF THE AUTOMORPHISM GROUPS OF SPORADIC SIMPLE GROUPS Behrooz Khosravi Abstract. In this paper as the main result, we determine finite
More informationHeights of characters and defect groups
[Page 1] Heights of characters and defect groups Alexander Moretó 1. Introduction An important result in ordinary character theory is the Ito-Michler theorem, which asserts that a prime p does not divide
More informationLANDAU S THEOREM, FIELDS OF VALUES FOR CHARACTERS, AND SOLVABLE GROUPS arxiv: v1 [math.gr] 26 Jun 2015
LANDAU S THEOREM, FIELDS OF VALUES FOR CHARACTERS, AND SOLVABLE GROUPS arxiv:1506.08169v1 [math.gr] 26 Jun 2015 MARK L. LEWIS Abstract. When G is solvable group, we prove that the number of conjugacy classes
More informationDiameter of the Zero Divisor Graph of Semiring of Matrices over Boolean Semiring
International Mathematical Forum, Vol. 9, 2014, no. 29, 1369-1375 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.47131 Diameter of the Zero Divisor Graph of Semiring of Matrices over
More informationGroups with Few Normalizer Subgroups
Irish Math. Soc. Bulletin 56 (2005), 103 113 103 Groups with Few Normalizer Subgroups FAUSTO DE MARI AND FRANCESCO DE GIOVANNI Dedicated to Martin L. Newell Abstract. The behaviour of normalizer subgroups
More informationFINITE GROUPS WITH FIVE NON-CENTRAL CONJUGACY CLASSES
Journal of Algebraic Systems Vol. 4, No. 2, (207), pp 85-95 DOI: 0.22044/jas.207.850 FINITE GROUPS WITH FIVE NON-CENTRAL CONJUGACY CLASSES M. REZAEI AND Z. FORUZANFAR Abstract. Let G be a finite group
More informationThe Class Equation X = Gx. x X/G
The Class Equation 9-9-2012 If X is a G-set, X is partitioned by the G-orbits. So if X is finite, X = x X/G ( x X/G means you should take one representative x from each orbit, and sum over the set of representatives.
More informationWeak Resolvable Spaces and. Decomposition of Continuity
Pure Mathematical Sciences, Vol. 6, 2017, no. 1, 19-28 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/pms.2017.61020 Weak Resolvable Spaces and Decomposition of Continuity Mustafa H. Hadi University
More informationz-classes in finite groups of conjugate type (n, 1)
Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:31 https://doi.org/10.1007/s12044-018-0412-5 z-classes in finite groups of conjugate type (n, 1) SHIVAM ARORA 1 and KRISHNENDU GONGOPADHYAY 2, 1 Department
More informationr-ideals of Commutative Semigroups
International Journal of Algebra, Vol. 10, 2016, no. 11, 525-533 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2016.61276 r-ideals of Commutative Semigroups Muhammet Ali Erbay Department of
More informationof a Two-Operator Product 1
Applied Mathematical Sciences, Vol. 7, 2013, no. 130, 6465-6474 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.39501 Reverse Order Law for {1, 3}-Inverse of a Two-Operator Product 1 XUE
More informationMappings of the Direct Product of B-algebras
International Journal of Algebra, Vol. 10, 2016, no. 3, 133-140 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.615 Mappings of the Direct Product of B-algebras Jacel Angeline V. Lingcong
More informationCHAPTER III NORMAL SERIES
CHAPTER III NORMAL SERIES 1. Normal Series A group is called simple if it has no nontrivial, proper, normal subgroups. The only abelian simple groups are cyclic groups of prime order, but some authors
More informationFuzzy Sequences in Metric Spaces
Int. Journal of Math. Analysis, Vol. 8, 2014, no. 15, 699-706 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.4262 Fuzzy Sequences in Metric Spaces M. Muthukumari Research scholar, V.O.C.
More informationNon Isolated Periodic Orbits of a Fixed Period for Quadratic Dynamical Systems
Applied Mathematical Sciences, Vol. 12, 2018, no. 22, 1053-1058 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.87100 Non Isolated Periodic Orbits of a Fixed Period for Quadratic Dynamical
More informationHUPPERT S CONJECTURE FOR F i 23
HUPPERT S CONJECTURE FOR F i 23 S. H. ALAVI, A. DANESHKAH, H. P. TONG-VIET, AND T. P. WAKEFIELD Abstract. Let G denote a finite group and cd(g) the set of irreducible character degrees of G. Bertram Huppert
More informationRepresentation of Semisimple Jordan and Lie Triple Systems
Applied Mathematical Sciences, Vol. 12, 2018, no. 8, 399-406 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.8117 Representation of Semisimple Jordan and Lie Triple Systems Ahmad Alghamdi
More informationHermitian Weighted Composition Operators on the Fock-type Space F 2 α(c N )
Applied Mathematical Sciences, Vol. 9, 2015, no. 61, 3037-3043 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.136 Hermitian Weighted Composition Operators on the Fock-type Space F 2 (C
More information1. Group Theory Permutations.
1.1. Permutations. 1. Group Theory Problem 1.1. Let G be a subgroup of S n of index 2. Show that G = A n. Problem 1.2. Find two elements of S 7 that have the same order but are not conjugate. Let π S 7
More informationPseudo Sylow numbers
Pseudo Sylow numbers Benjamin Sambale May 16, 2018 Abstract One part of Sylow s famous theorem in group theory states that the number of Sylow p- subgroups of a finite group is always congruent to 1 modulo
More informationExtending Brauer s Height Zero Conjecture to blocks with nonabelian defect groups
Extending Brauer s Height Zero Conjecture to blocks with nonabelian defect groups Charles W. Eaton and Alexander Moretó Abstract We propose a generalization of Brauer s Height Zero Conjecture that considers
More informationCONSEQUENCES OF THE SYLOW THEOREMS
CONSEQUENCES OF THE SYLOW THEOREMS KEITH CONRAD For a group theorist, Sylow s Theorem is such a basic tool, and so fundamental, that it is used almost without thinking, like breathing. Geoff Robinson 1.
More informationGeneralized Derivation on TM Algebras
International Journal of Algebra, Vol. 7, 2013, no. 6, 251-258 HIKARI Ltd, www.m-hikari.com Generalized Derivation on TM Algebras T. Ganeshkumar Department of Mathematics M.S.S. Wakf Board College Madurai-625020,
More informationConvex Sets Strict Separation in Hilbert Spaces
Applied Mathematical Sciences, Vol. 8, 2014, no. 64, 3155-3160 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.44257 Convex Sets Strict Separation in Hilbert Spaces M. A. M. Ferreira 1
More informationSkew Cyclic and Quasi-Cyclic Codes of Arbitrary Length over Galois Rings
International Journal of Algebra, Vol. 7, 2013, no. 17, 803-807 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2013.310100 Skew Cyclic and Quasi-Cyclic Codes of Arbitrary Length over Galois
More information