Supersolubility of Finite Groups 1
|
|
- Wendy Lane
- 5 years ago
- Views:
Transcription
1 Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 20, Completely c-semipermutable Subgroups and Supersolubility of Finite Groups 1 Bin Hu 1, Jianhong Huang 1,2 and Lifang Ma 1 1. Department of Mathematics, Xuzhou Normal University Xuzhou, , P.R. China 2. Department of Mathematics University of Science and Technology of China Hefei , P.R. China hubin118@126.com Abstract A subgroup A of a group G is said to be completely conditionally semipermutable ( or in brevity, completely c-semipermutable )in G if A has a minimal supplement T in G such that for every subgroup T 1 of T there exists an element x T A, T 1 such that AT1 x = T 1 x A. In this paper, we use completely c-semipermutable subgroups to study some new criteria of supersolubility for products of finite groups. Mathematics Subject Classification: 20D10, 20D15, 20D20 Keywords: Finite group, Sylow subgroup, completely c-semipermutable subgroup 1. Introduction Throughout this paper, all groups considered are finite and G denotes a group. The terminology and notation are standard, as in [1] and [2]. 1 Research is supported by NNSF Grant of China(grant: )and Project supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No.10KJD110004).
2 986 Bin Hu, Jianhong Huang and Lifang Ma Recall that a subgroup A of a group G is called permutable with a subgroup B if AB = BA. IfA is permutable with all subgroups of G, then A is called a permutable subgroup or a quasinormal subgroup of G [1, 8]. The permutable subgroups have many interesting properties. For example, Ore [9] proved that every permutable subgroup of a finite group is subnormal. Ito and Szép [8] have already proved that if a subgroup H is quasinormal in G, then H/H G is nilpotent. However, in general, two subgroups H and T of G may not be permutable, but G may contain an element x such that HT x = T x H. Recently, Guo, Shum and Skiba[4] introduced the concept of conditionally permutable subgroup (generally, the concept of H-permutable subgroup). Using this new concept, Guo, Shum and Skiba have obtained some new elegant results on the structure of finite groups [3-5]. As a development of the results above, we introduced the concept of completely c-semipermutable subgroups and have used it to determine the structure of some finite groups [6-7]. Definition 1.1 [6]. A subgroup A of a group G is said to be completely conditionally semipermutable (or in brevity, completely c-semipermutable)in G if A has a minimal supplement T in G such that for every subgroup T 1 of T there exists an element x T A, T 1 such that AT1 x = T 1 xa. In this paper, we apply completely c-semipermutable subgroups to study some new criteria of supersolubility for products of finite groups. 2. Preliminary results For the reader s convenience, we cite some known results which are useful in the sequel. Lemma 2.1 [6, Lemma 2.3]. Let G be a group, N G and A G. Then the following statements hold: (1) If A is completely c-semipermutable in G, then AN/N is completely c-semipermutable in G/N. (2) If A/N is completely c-semipermutable in G/N, then A is completely c-semipermutable in G. (3) If A is completely c-semipermutable in G and A H G, then A is completely c-semipermutable in H.
3 Completely c-semipermutable subgroups 987 (4) If T T c (A), then T x T c (A) for all x G. Lemma 2.2 [7]. A group G is supersoluble if all Sylow subgroups of G are completely c-semipermutable in G. 3. Main results Theorem 3.1. Let G = AB be the product of subgroups A and B and ( A, B ) =1. If every Sylow subgroup of A and B is completely c-semipermutable in G, then G is supersoluble. Proof. Assume that the theorem is false and let G be a counterexample of minimal order. Then: (1) G/N is supersoluble for every non-identity normal subgroup N of G. Let 1 N G. Then G/N =(AN/N)(BN/N) and AN/N A/(A N), BN/N B/(B N). Since ( A, B ) = 1, ( AN/N, BN/N ) = 1. Let r be an arbitrary prime divisor of AN/N and T/N be a Sylow r-subgroup of AN/N. Then there exists a Sylow r-subgroup R of A such that T = RN. By hypothesis and Lemma 2.1(1), we have that T/N is completely c- semipermutable in G/N. By the same discussion, every Sylow subgroup of BN/N is completely c-semipermutable in G/N. Therefore, G/N satisfies the hypothesis. By the choice of G, G/N is supersoluble. (2) G is soluble. By Lemma 2.1(3) and Lemma 2.2, both A and B are supersoluble. Let p be the largest prime divisor of G. Since ( A, B ) = 1, without loss of generality, we can suppose that p π(a). Assume that P is a Sylow p-subgroup of A. Then P A. Obviously, P is also a Sylow p-subgroup of G. By hypothesis, P is completely c-semipermutable in G. So there is a minimal supplement T in G such that G = PT. Let q be any prime divisor of G with q p. Then there exists a Sylow q-subgroup Q of G such that PQ = QP. Assume that q π(b). We can see that B has a Sylow q-subgroup Q 1 such that Q t 1 = Q, where t G. Hence PQt 1 = Qt 1P. Since t G and G = AB, t = ab, where a A, b B. It follows that PQ ab 1 = Qab 1 P and so PQb 1 = Qb 1 P. Hence, PQ b 1 satisfies the hypothesis. If G = PQ b 1, then G is soluble. Assume that PQ b 1 <G. By the choice of G, PQ b 1 is supersoluble and so Q b 1 N G (P ). By the arbitrary choice of q, we have that P G. In view of (1), G is soluble.
4 988 Bin Hu, Jianhong Huang and Lifang Ma (3) G has the unique minimal normal subgroup N and N = O p (G) = F (G) =C G (N), for some prime p. Since the class of all supersoluble groups is a saturated formation, G has the unique minimal normal subgroup N. It is easy to see that N = O p (G) = F (G) =C G (N) for G is soluble. (4) Final contradiction. Let Q be an arbitrary Sylow q-subgroup of A or B with q p. by hypothesis, Q is completely c-semipermutable in G. So there is a minimal supplement T in G such that G = QT. Let P be a Sylow p-subgroup of T. Then N P. Let 1 x N Z(P ). Then Q t x = x Q t, where t T Q, x. Therefore, x = x (Q t N) = x Q t N x Q t. It follows that Q t N G ( x ). By Lemma 2.1(4), we have that every Sylow q-subgroup of A or B has a minimal supplement T in G such that x T. Hence, x G by the arbitrary choice of q. Since N is the unique minimal normal subgroup of G, N = x. By (1), G is supersoluble. The contradiction completes the proof. Theorem 3.2. Let G be a group and G = AB, where A and B are the square-free subgroups of G. If every Sylow subgroup of A and B is completely c-semipermutable in G, then G is supersoluble. Proof. Assume that the theorem is false and let G be a counterexample of minimal order. Then: (1) G/N is supersoluble for every non-trivial normal subgroup N of G. Obviously, G/N =(AN/N)(BN/N). Since AN/N A/A N,BN/N B/B N, AN/N and BN/N are the square-free groups. Let T/N be any Sylow p-subgroup of AN/N. Then there exists a Sylow p-subgroup P of A such that T = PN. By hypothesis and Lemma 2.1(1), T/N is completely c-semipermutable in G/N. By the same discussion, every Sylow subgroup of BN/N is completely c-semipermutable in G/N. Hence G/N satisfies the hypothesis. By the choice of G, G/N is supersoluble. (2) G is soluble and G has the unique minimal normal subgroup N. By [10, IX, Lemma 4.7], we have that G is soluble. Since the class of all supersoluble groups is a saturated formation, G has the unique minimal normal subgroup N. (3) N = p 2, where p is a prime divisor of G. By hypothesis, N = p or N = p 2, where p is a prime divisor of G.
5 Completely c-semipermutable subgroups 989 If N = p, then by (1), G is supersoluble. This contradiction shows that N = p 2 and consequently N is the Sylow p-subgroup of G. (4) Final contradiction. Let P = x be an arbitrary Sylow p-subgroup of A or B. obviously, P N. By hypothesis, P is completely c-semipermutable in G and so there exists a minimal supplement T of G such that G = PT = x T. For any prime r p, there exists a Sylow r-subgroup R of G such that R x = x R. Since x = x (R N) = x R N x R, R N G ( x ). On the other hand, since N is abelian, we have that x N. Hence, G = N G ( x ) and x G. By (1), G is supersoluble. This contradiction completes the proof. References [1] K.Doerk and T.Hawkes, Finite soluble groups, Walter de gruyter, Berlin/New York, [2] W.Guo, The Theory of Class of Groups, Science Press-Kluwer Academic Publishers, Beijing-New-York-Dordrecht-Boston, [3] W.Guo, K.P.Shum and A.Skiba, Conditionally Permutable subgroups and Supersolubility of Finite Groups, SEAMS Bull Math., 29(2005), [4] W.Guo, K.P.Shum and A.Skiba, G-covering systems of subgroups for classes of p-supersoluble and p-nilpotent of finite groups, Siberian Mathematical Journal, 45(2004), [5] W.Guo, K.P.Shum and A.Skiba, Criterions of supersolubility for products of supersoluble groups, Publications Math. Debreceen, 68(2005), [6] B.Hu, W.Guo, c-semipermutable Subgroups of Finite Groups, Siberian Mathematical Journal, 48(2007), [7] B.Hu, J.Huang and N.Yang, Completely c-semipermutable Subgroups of Finite Groups, Math.Sci.Res.J., 10(2006), [8] N.Ito, J.Szép, Uber die quasinormalteiler endlicher gruppen, Act. Sci. Math., 23(1962),
6 990 Bin Hu, Jianhong Huang and Lifang Ma [9] O.Ore, Contributions in the theorey of groups of finite order, Duke Math. J., 5(1939), [10] M.Xu, An Introduction to Finite Groups, Science Press, Beijing, Received: October, 2010
The 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 informationOn partially τ-quasinormal subgroups of finite groups
Hacettepe Journal of Mathematics and Statistics Volume 43 (6) (2014), 953 961 On partially τ-quasinormal subgroups of finite groups Changwen Li, Xuemei Zhang and Xiaolan Yi Abstract Let H be a subgroup
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 informationActa Mathematica Academiae Paedagogicae Nyíregyháziensis 24 (2008), ISSN
Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 24 (2008), 297 307 www.emis.de/journals ISSN 1786-0091 ON S-QUASINORMAL SUBGROUPS OF FINITE GROUP JEHAD J. JARADEN Abstract. A subgroup H of a group
More informationSome New Criteria on the Supersolvability of Finite Groups
Int. J. Pure Appl. Sci. Technol., 5(2) (20), pp. 9-30 International Journal of Pure and Applied Sciences and Technology ISSN 2229-607 Available online at www.ijopaasat.in Research Paper Some New Criteria
More informationCyclic non-s-permutable subgroups and non-normal maximal subgroups
Rend. Sem. Mat. Univ. Padova 1xx (201x) Rendiconti del Seminario Matematico della Università di Padova c European Mathematical Society Cyclic non-s-permutable subgroups and non-normal maximal subgroups
More informationA NOTE ON PRIMITIVE SUBGROUPS OF FINITE SOLVABLE GROUPS
Commun. Korean Math. Soc. 28 (2013), No. 1, pp. 55 62 http://dx.doi.org/10.4134/ckms.2013.28.1.055 A NOTE ON PRIMITIVE SUBGROUPS OF FINITE SOLVABLE GROUPS Xuanli He, Shouhong Qiao, and Yanming Wang Abstract.
More informationOn W -S-permutable Subgroups of Finite Groups
Rend. Sem. Mat. Univ. Padova 1xx (201x) Rendiconti del Seminario Matematico della Università di Padova c European Mathematical Society On W -S-permutable Subgroups of Finite Groups Jinxin Gao Xiuyun Guo
More informationMath-Net.Ru All Russian mathematical portal
Math-Net.Ru All Russian mathematical portal Wenbin Guo, Alexander N. Skiba, On the intersection of maximal supersoluble subgroups of a finite group, Tr. Inst. Mat., 2013, Volume 21, Number 1, 48 51 Use
More informationON SEMINORMAL SUBGROUPS OF FINITE GROUPS
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 47, Number 2, 2017 ON SEMINORMAL SUBGROUPS OF FINITE GROUPS A. BALLESTER-BOLINCHES, J.C. BEIDLEMAN, V. PÉREZ-CALABUIG AND M.F. RAGLAND ABSTRACT. All groups
More informationJournal of Algebra. Radicals in mutually permutable products of finite groups
Journal of Algebra 321 (2009) 353 360 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra Radicals in mutually permutable products of finite groups Jonas Bochtler
More informationON CONDITIONAL PERMUTABILITY AND SATURATED FORMATIONS
Proceedings of the Edinburgh Mathematical Society (2011) 54, 309 319 DOI:10.1017/S0013091510000106 ON CONDITIONAL PERMUTABILITY AND SATURATED FORMATIONS M. ARROYO-JORDÁ1, P. ARROYO-JORDÁ1 AND M. D. PÉREZ-RAMOS2
More informationThe influence of C- Z-permutable subgroups on the structure of finite groups
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 1, 2018, 160-168 ISSN 1307-5543 www.ejpam.com Published by New York Business Global The influence of C- Z-permutable subgroups on the structure
More informationThe final publication is available at
Document downloaded from: http://hdl.handle.net/0/ This paper must be cited as: Ballester Bolinches, A.; Beidleman, JC.; Cossey, J.; Esteban Romero, R.; Rangland, MF.; Schmidt, J. (00). Permutable subnormal
More informationNew progress on factorized groups and subgroup permutability
New progress on factorized groups and subgroup permutability Paz Arroyo-Jordá Instituto Universitario de Matemática Pura y Aplicada Universidad Politécnica de Valencia, Spain Groups St Andrews 2013 in
More informationGROUPS WITH PERMUTABILITY CONDITIONS FOR SUBGROUPS OF INFINITE RANK. Communicated by Patrizia Longobardi. 1. Introduction
International Journal of Group Theory ISSN (print): 2251-7650, ISSN (on-line): 2251-7669 Vol. 7 No. 3 (2018), pp. 7-16. c 2018 University of Isfahan ijgt.ui.ac.ir www.ui.ac.ir GROUPS WITH PERMUTABILITY
More informationDepartment of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box: 80111, Jeddah 21589, Saudi Arabia.
On Product Groups in which S-semipermutability Is atransitive Relation Mustafa Obaid Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box: 80111, Jeddah 21589, Saudi Arabia.
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 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 information2-GENERATED CAYLEY DIGRAPHS ON NILPOTENT GROUPS HAVE HAMILTONIAN PATHS
Volume 7, Number 1, Pages 41 47 ISSN 1715-0868 2-GENERATED CAYLEY DIGRAPHS ON NILPOTENT GROUPS HAVE HAMILTONIAN PATHS DAVE WITTE MORRIS Abstract. Suppose G is a nilpotent, finite group. We show that if
More informationThis paper has been published in Journal of Algebra, 319(8): (2008).
This paper has been published in Journal of Algebra, 319(8):3343 3351 (2008). Copyright 2008 by Elsevier. The final publication is available at www.sciencedirect.com. http://dx.doi.org/10.1016/j.jalgebra.2007.12.001
More informationSOME TOPICS ON PERMUTABLE SUBGROUPS IN INFINITE GROUPS
SOME TOPICS ON PERMUTABLE SUBGROUPS IN INFINITE GROUPS TESI DI DOTTORATO SCIENZE MATEMATICHE E INFORMATICHE - XXIX CICLO DIPARTIMENTO DI MATEMATICA E APPLICAZIONI RENATO CACCIOPPOLI ROBERTO IALENTI UNIVERSITÀ
More informationOn CAP-quasinormal Subgroups of Finite Groups
Λ46ΩΛ6fl Ψ Vol. 46, No. 6 2017fi11» ADVANCES IN MATHEMATICS (CHINA) Nov., 2017 On CAP-quasinormal Subgroups of Finite Groups ZHANG Li, WU Zhenfeng, GUO Wenbin doi: 10.11845/sxjz.2016032b (School of Mathematical
More informationOn Finite Groups in which Every Solvable Non-cyclic Proper Subgroup is either Self-normalizing or Normal 1
International Journal of Algebra, Vol. 6, 2012, no. 23, 1111-1115 On Finite Groups in which Every Solvable Non-cyclic Proper Subgroup is either Self-normalizing or Normal 1 Shuxia Zhang School of Mathematics
More informationThe Structure of Minimal Non-ST-Groups
2012 2nd International Conference on Industrial Technology and Management (ICITM 2012) IPCSIT vol. 49 (2012) (2012) IACSIT Press, Singapore DOI: 10.7763/IPCSIT.2012.V49.41 The Structure of Minimal Non-ST-Groups
More informationCompact Primitive Semigroups Having (CEP)
International Journal of Algebra, Vol. 3, 2009, no. 19, 903-910 Compact Primitive Semigroups Having (CEP) Xiaojiang Guo 1 Department of Mathematics, Jiangxi Normal University Nanchang, Jiangxi 330022,
More informationFinite groups determined by an inequality of the orders of their subgroups
Finite groups determined by an inequality of the orders of their subgroups Tom De Medts Marius Tărnăuceanu November 5, 2007 Abstract In this article we introduce and study two classes of finite groups
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 informationFINITE GROUPS WHOSE SUBNORMAL SUBGROUPS PERMUTE WITH ALL SYLOW SUBGROUPS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 47, Number 1, January 1975 FINITE GROUPS WHOSE SUBNORMAL SUBGROUPS PERMUTE WITH ALL SYLOW SUBGROUPS RAM K. AGRAWAL1 ABSTRACT. As a generalization
More informationON THE SUM OF ELEMENT ORDERS OF FINITE ABELIAN GROUPS
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI (S.N.) MATEMATICĂ, Tomul...,..., f... DOI: 10.2478/aicu-2013-0013 ON THE SUM OF ELEMENT ORDERS OF FINITE ABELIAN GROUPS BY MARIUS TĂRNĂUCEANU and
More informationOn Nearly S-Permutably Embedded Subgroups of Finite Groups *
ISSN 0001-4346, Mathematical Notes, 2012, Vol. 91, No. 4, pp. 470 478. Pleiades Publishing, Ltd., 2012. Published in Russian in Matematicheskie Zametki, 2012, Vol. 91, No. 4, pp. 495 505. On Nearly S-Permutably
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 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 informationArc-transitive pentavalent graphs of order 4pq
Arc-transitive pentavalent graphs of order 4pq Jiangmin Pan Bengong Lou Cuifeng Liu School of Mathematics and Statistics Yunnan University Kunming, Yunnan, 650031, P.R. China Submitted: May 22, 2012; Accepted:
More informationPERMUTABILITY GRAPH OF CYCLIC SUBGROUPS
Palestine Journal of Mathematics Vol. 5(2) (2016), 228 239 Palestine Polytechnic University-PPU 2016 PERMUTABILITY GRAPH OF CYCLIC SUBGROUPS R. Rajkumar and P. Devi Communicated by Ayman Badawi MSC 2010
More informationarxiv: v1 [math.gr] 6 Nov 2017
On classes of finite groups with simple non-abelian chief factors V. I. Murashka {mvimath@yandex.ru} Francisk Skorina Gomel State University, Gomel, Belarus arxiv:1711.01686v1 [math.gr] 6 Nov 2017 Abstract.
More informationChief factors. Jack Schmidt. University of Kentucky
Chief factors Jack Schmidt University of Kentucky 2008-03-05 Chief factors allow a group to be studied by its representation theory on particularly natural irreducible modules. Outline What is a chief
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 informationSection II.8. Normal and Subnormal Series
II.8. Normal and Subnormal Series 1 Section II.8. Normal and Subnormal Series Note. In this section, two more series of a group are introduced. These will be useful in the Insolvability of the Quintic.
More informationCHARACTERIZATION OF PROJECTIVE GENERAL LINEAR GROUPS. Communicated by Engeny Vdovin. 1. Introduction
International Journal of Group Theory ISSN (print): 51-7650, ISSN (on-line): 51-7669 Vol. 5 No. 1 (016), pp. 17-8. c 016 University of Isfahan www.theoryofgroups.ir www.ui.ac.ir CHARACTERIZATION OF PROJECTIVE
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 informationRecognising nilpotent groups
Recognising nilpotent groups A. R. Camina and R. D. Camina School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK; a.camina@uea.ac.uk Fitzwilliam College, Cambridge, CB3 0DG, UK; R.D.Camina@dpmms.cam.ac.uk
More informationSOME NEW PERMUTABILITY PROPERTIES OF HYPERCENTRALLY EMBEDDED SUBGROUPS OF FINITE GROUPS
J. Aust. Math. Soc. 79 (2005), 243 255 SOME NEW PERMUTABILITY PROPERTIES OF HYPERCENTRALLY EMBEDDED SUBGROUPS OF FINITE GROUPS L. M. EZQUERRO and X. SOLER-ESCRIVÀ (Received 19 December 2003; revised 10
More informationNILPOTENT NUMBERS JONATHAN PAKIANATHAN AND KRISHNAN SHANKAR
NILPOTENT NUMBERS JONATHAN PAKIANATHAN AND KRISHNAN SHANKAR Introduction. One of the first things we learn in abstract algebra is the notion of a cyclic group. For every positive integer n, we have Z n,
More informationSection II.2. Finitely Generated Abelian Groups
II.2. Finitely Generated Abelian Groups 1 Section II.2. Finitely Generated Abelian Groups Note. In this section we prove the Fundamental Theorem of Finitely Generated Abelian Groups. Recall that every
More information3. G. Groups, as men, will be known by their actions. - Guillermo Moreno
3.1. The denition. 3. G Groups, as men, will be known by their actions. - Guillermo Moreno D 3.1. An action of a group G on a set X is a function from : G X! X such that the following hold for all g, h
More informationThe influence of order and conjugacy class length on the structure of finite groups
Hokkaido Mathematical Journal Vol. 47 (2018) p. 25 32 The influence of order and conjugacy class length on the structure of finite groups Alireza Khalili Asboei, Mohammad Reza Darafsheh and Reza Mohammadyari
More informationFINITE GROUPS IN WHICH SOME PROPERTY OF TWO-GENERATOR SUBGROUPS IS TRANSITIVE
FINITE GROUPS IN WHICH SOME PROPERTY OF TWO-GENERATOR SUBGROUPS IS TRANSITIVE COSTANTINO DELIZIA, PRIMOŽ MORAVEC, AND CHIARA NICOTERA Abstract. Finite groups in which a given property of two-generator
More informationGROUPS IN WHICH SYLOW SUBGROUPS AND SUBNORMAL SUBGROUPS PERMUTE
Illinois Journal of Mathematics Volume 47, Number 1/2, Spring/Summer 2003, Pages 63 69 S 0019-2082 GROUPS IN WHICH SYLOW SUBGROUPS AND SUBNORMAL SUBGROUPS PERMUTE A. BALLESTER-BOLINCHES, J. C. BEIDLEMAN,
More informationMath 430 Exam 2, Fall 2008
Do not distribute. IIT Dept. Applied Mathematics, February 16, 2009 1 PRINT Last name: Signature: First name: Student ID: Math 430 Exam 2, Fall 2008 These theorems may be cited at any time during the test
More informationFINITE GROUPS WHICH ARE PRODUCTS OF PAIRWISE TOTALLY PERMUTABLE SUBGROUPS
Proceedings of the Edinburgh Mathematical Society (1998) 41, 567-572 FINITE GROUPS WHICH ARE PRODUCTS OF PAIRWISE TOTALLY PERMUTABLE SUBGROUPS by A. BALLESTER-BOLINCHES, M. C. PEDRAZA-AGUILERA and M. D.
More informationMINIMAL NUMBER OF GENERATORS AND MINIMUM ORDER OF A NON-ABELIAN GROUP WHOSE ELEMENTS COMMUTE WITH THEIR ENDOMORPHIC IMAGES
Communications in Algebra, 36: 1976 1987, 2008 Copyright Taylor & Francis roup, LLC ISSN: 0092-7872 print/1532-4125 online DOI: 10.1080/00927870801941903 MINIMAL NUMBER OF ENERATORS AND MINIMUM ORDER OF
More informationA characterisation of p-soluble groups
A characterisation of p-soluble groups Paul Flavell The School of Mathematics and Statistics The University of Birmingham Birmingham B15 2TT United Kingdom e-mail: p.j.flavell@bham.ac.uk If p is a prime
More informationOdd-order Cayley graphs with commutator subgroup of order pq are hamiltonian
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 1 28 Odd-order Cayley graphs with commutator subgroup of order
More informationGraphs and Classes of Finite Groups
Note di Matematica ISSN 1123-2536, e-issn 1590-0932 Note Mat. 33 (2013) no. 1, 89 94. doi:10.1285/i15900932v33n1p89 Graphs and Classes of Finite Groups A. Ballester-Bolinches Departament d Àlgebra, Universitat
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 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 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 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 informationPacific Journal of Mathematics
Pacific Journal of Mathematics SOME PROPERTIES OF THE PROBABILISTIC ZETA FUNCTION OF FINITE SIMPLE GROUPS Erika Damian, Andrea Lucchini, and Fiorenza Morini Volume 215 No. 1 May 2004 PACIFIC JOURNAL OF
More informationPrime and irreducible elements of the ring of integers modulo n
Prime and irreducible elements of the ring of integers modulo n M. H. Jafari and A. R. Madadi Department of Pure Mathematics, Faculty of Mathematical Sciences University of Tabriz, Tabriz, Iran Abstract
More informationA SIMPLE PROOF OF BURNSIDE S CRITERION FOR ALL GROUPS OF ORDER n TO BE CYCLIC
A SIMPLE PROOF OF BURNSIDE S CRITERION FOR ALL GROUPS OF ORDER n TO BE CYCLIC SIDDHI PATHAK Abstract. This note gives a simple proof of a famous theorem of Burnside, namely, all groups of order n are cyclic
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 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 informationMath 250A, Fall 2004 Problems due October 5, 2004 The problems this week were from Lang s Algebra, Chapter I.
Math 250A, Fall 2004 Problems due October 5, 2004 The problems this week were from Lang s Algebra, Chapter I. 24. We basically know already that groups of order p 2 are abelian. Indeed, p-groups have non-trivial
More informationGROUPS WITH A MAXIMAL IRREDUNDANT 6-COVER #
Communications in Algebra, 33: 3225 3238, 2005 Copyright Taylor & Francis, Inc. ISSN: 0092-7872 print/1532-4125 online DOI: 10.1081/AB-200066157 ROUPS WITH A MAXIMAL IRREDUNDANT 6-COVER # A. Abdollahi,
More informationAn Abundancy Result for the Two Prime Power Case and Results for an Equations of Goormaghtigh
International Mathematical Forum, Vol. 8, 2013, no. 9, 427-432 HIKARI Ltd, www.m-hikari.com An Abundancy Result for the Two Prime Power Case and Results for an Equations of Goormaghtigh Richard F. Ryan
More informationA Note on Subgroup Coverings of Finite Groups
Analele Universităţii de Vest, Timişoara Seria Matematică Informatică XLIX, 2, (2011), 129 135 A Note on Subgroup Coverings of Finite Groups Marius Tărnăuceanu Abstract. In this note we determine the finite
More informationGroups with many subnormal subgroups. *
Groups with many subnormal subgroups. * Eloisa Detomi Dipartimento di Matematica, Facoltà di Ingegneria, Università degli Studi di Brescia, via Valotti 9, 25133 Brescia, Italy. E-mail: detomi@ing.unibs.it
More informationGroups that are pairwise nilpotent
Groups that are pairwise nilpotent Gérard Endimioni C.M.I, Université de Provence, UMR-CNRS 6632 39, rue F. Joliot-Curie, 13453 Marseille Cedex 13, France E-mail: endimion@gyptis.univ-mrs.fr Gunnar Traustason
More informationActa Arith., 140(2009), no. 4, ON BIALOSTOCKI S CONJECTURE FOR ZERO-SUM SEQUENCES. 1. Introduction
Acta Arith., 140(009), no. 4, 39 334. ON BIALOSTOCKI S CONJECTURE FOR ZERO-SUM SEQUENCES SONG GUO AND ZHI-WEI SUN* arxiv:081.174v3 [math.co] 16 Dec 009 Abstract. Let n be a positive even integer, and let
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 informationA conjugacy criterion for Hall subgroups in finite groups
MSC2010 20D20, 20E34 A conjugacy criterion for Hall subgroups in finite groups E.P. Vdovin, D.O. Revin arxiv:1004.1245v1 [math.gr] 8 Apr 2010 October 31, 2018 Abstract A finite group G is said to satisfy
More informationCharacterization of the Linear Groups PSL(2, p)
Southeast Asian Bulletin of Mathematics (2014) 38: 471 478 Southeast Asian Bulletin of Mathematics c SEAMS. 2014 Characterization of the Linear Groups PSL(2, p) Alireza Khalili Asboei Department of Mathematics,
More informationNORMAL SMASH PRODUCTS
PORTUGALIAE MATHEMATICA Vol. 55 Fasc. 3 1998 NORMAL SMASH PRODUCTS S. Yang and D. Wang* Abstract: Let H be a co-frobenius Hopf algebra over a field k and A a right H-comodule algebra. It is shown that
More informationCheck Character Systems Using Chevalley Groups
Designs, Codes and Cryptography, 10, 137 143 (1997) c 1997 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Check Character Systems Using Chevalley Groups CLAUDIA BROECKER 2. Mathematisches
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 informationOn the prime graph of L 2 (q) where q = p α < Introduction
Quasigroups and Related Systems 14 (2006), 179 190 On the prime graph of L 2 (q) where q = p α < 100 Behrooz Khosravi and Seyyed Sadegh Salehi Amiri Abstract Let G be a nite group. We construct the prime
More informationA generalization of the Euler s totient function
arxiv:1312.1428v1 [math.gr] 5 Dec 2013 A generalization of the Euler s totient function Marius Tărnăuceanu December 5, 2013 Abstract The main goal of this paper is to provide a group theoretical generalization
More informationSECOND-ORDER RECURRENCES. Lawrence Somer Department of Mathematics, Catholic University of America, Washington, D.C
p-stability OF DEGENERATE SECOND-ORDER RECURRENCES Lawrence Somer Department of Mathematics, Catholic University of America, Washington, D.C. 20064 Walter Carlip Department of Mathematics and Computer
More informationSubrings and Ideals 2.1 INTRODUCTION 2.2 SUBRING
Subrings and Ideals Chapter 2 2.1 INTRODUCTION In this chapter, we discuss, subrings, sub fields. Ideals and quotient ring. We begin our study by defining a subring. If (R, +, ) is a ring and S is a non-empty
More informationAn arithmetic theorem related to groups of bounded nilpotency class
Journal of Algebra 300 (2006) 10 15 www.elsevier.com/locate/algebra An arithmetic theorem related to groups of bounded nilpotency class Thomas W. Müller School of Mathematical Sciences, Queen Mary & Westfield
More informationC-Characteristically Simple Groups
BULLETIN of the Malaysian Mathematical Sciences Society http://math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. (2) 35(1) (2012), 147 154 C-Characteristically Simple Groups M. Shabani Attar Department
More informationarxiv: v2 [math.gr] 17 Dec 2017
The complement of proper power graphs of finite groups T. Anitha, R. Rajkumar arxiv:1601.03683v2 [math.gr] 17 Dec 2017 Department of Mathematics, The Gandhigram Rural Institute Deemed to be University,
More informationProblem 1.1. Classify all groups of order 385 up to isomorphism.
Math 504: Modern Algebra, Fall Quarter 2017 Jarod Alper Midterm Solutions Problem 1.1. Classify all groups of order 385 up to isomorphism. Solution: Let G be a group of order 385. Factor 385 as 385 = 5
More informationLECTURES 11-13: CAUCHY S THEOREM AND THE SYLOW THEOREMS
LECTURES 11-13: CAUCHY S THEOREM AND THE SYLOW THEOREMS Recall Lagrange s theorem says that for any finite group G, if H G, then H divides G. In these lectures we will be interested in establishing certain
More informationOn the structure of some modules over generalized soluble groups
On the structure of some modules over generalized soluble groups L.A. Kurdachenko, I.Ya. Subbotin and V.A. Chepurdya Abstract. Let R be a ring and G a group. An R-module A is said to be artinian-by-(finite
More information1.5 Applications Of The Sylow Theorems
14 CHAPTER1. GROUP THEORY 8. The Sylow theorems are about subgroups whose order is a power of a prime p. Here is a result about subgroups of index p. Let H be a subgroup of the finite group G, and assume
More informationNormally monomial p-groups
Journal of Algebra 300 (2006) 2 9 www.elsevier.com/locate/jalgebra Normally monomial p-groups Avinoam Mann Einstein Institute of Mathematics, Hebrew University, Jerusalem, Israel Received 30 May 2005 Available
More informationNONABELIAN GROUPS WITH PERFECT ORDER SUBSETS
NONABELIAN GROUPS WITH PERFECT ORDER SUBSETS CARRIE E. FINCH AND LENNY JONES Abstract. Let G be a finite group and let x G. Define the order subset of G determined by x to be the set of all elements in
More informationInternational Journal Of Scientific & Engineering Research, Volume 7, Issue 7, July-2016 ISSN
1510 A Cayley graphs for Symmtric group on Degree four S.V IJAY AKUMAR C.V.R.HARINARAY ANAN Research Scholar Department of Mathematics PRIST University ThanjavurTamilnadu India. Research Supervisor Assistant
More information-ARC-TRANSITIVE GRAPHS Dragan Marusic IMFM, Oddelek za matematiko Univerza v Ljubljani Jadranska 19, 1111 Ljubljana Slovenija
CLASSIFYING -ARC-TRANSITIVE GRAPHS OF ORDER A PRODUCT OF TWO PRIMES Dragan Marusic 1 Primoz Potocnik 1 IMFM, Oddelek za matematiko IMFM, Oddelek za matematiko Univerza v Ljubljani Univerza v Ljubljani
More informationA HALL-TYPE CLOSURE PROPERTY FOR CERTAIN FITTING CLASSES
/. Austral. Math. Soc. (Series A) 59 (1995), 204-213 A HALL-TYPE CLOSURE PROPERTY FOR CERTAIN FITTING CLASSES OWEN J. BRISON (Received 27 July 1991; revised 4 February 1993) Communicated by H. Lausch Abstract
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 informationISOMORPHISM OF COMMUTATIVE MODULAR GROUP ALGEBRAS. P.V. Danchev
Serdica Math. J. 23 (1997), 211-224 ISOMORPHISM OF COMMUTATIVE MODULAR GROUP ALGEBRAS P.V. Danchev Communicated by L.L. Avramov Abstract. Let K be a field of characteristic p > 0 and let G be a direct
More informationSchool of Mathematics and Statistics. MT5824 Topics in Groups. Problem Sheet I: Revision and Re-Activation
MRQ 2009 School of Mathematics and Statistics MT5824 Topics in Groups Problem Sheet I: Revision and Re-Activation 1. Let H and K be subgroups of a group G. Define HK = {hk h H, k K }. (a) Show that HK
More informationD. S. Passman. University of Wisconsin-Madison
ULTRAPRODUCTS AND THE GROUP RING SEMIPRIMITIVITY PROBLEM D. S. Passman University of Wisconsin-Madison Abstract. This expository paper is a slightly expanded version of the final talk I gave at the group
More informationOn Torsion-by-Nilpotent Groups
On Torsion-by-Nilpotent roups. Endimioni C.M.I. - Université de Provence, UMR-CNRS 6632, 39, rue F. Joliot-Curie, 13453 Marseille Cedex 13, France E-mail: endimion@gyptis.univ-mrs.fr and. Traustason Department
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 informationABELIAN GROUPS WHOSE SUBGROUP LATTICE IS THE UNION OF TWO INTERVALS
J. Aust. Math. Soc. 78 (2005), 27 36 ABELIAN GROUPS WHOSE SUBGROUP LATTICE IS THE UNION OF TWO INTERVALS SIMION BREAZ and GRIGORE CĂLUGĂREANU (Received 15 February 2003; revised 21 July 2003) Communicated
More information