א א.. א מ..מ א! "......#$ % &'.. ( א 2

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "א א.. א מ..מ א! "......#$ % &'.. ( א 2"

Transcription

1 / - 1

2 א א אמ מא א 2

3 א א אאאאאא אאאא א א אאאא אמא אאאא א מאאאאאא אמאאא מ אאמאאא אאמ 3

4 א א 4

5 : - : : - 5

6 ( alois theory) ( ) Joseph Louis Lagrange ( ) Augustine Louis Cauchy Ludwid Sylow ( ) Camille Jordan Milter Shur Burnside Helpert Frabinous 6

7 (Class of finite super soluble groups) (Class of abelian finite groups) (Class of finite generated nilpotent groups) (Composition series) (Central series) (Normal series ) (Subnormal series) (Soluble groups) (Nilpotent groups) (Polycyclic groups) (Super soluble groups) 7

8 מאאאא א 8

9 / / Introduction to group theory W LEDERMANN // 9

10 , H, x, y, z, x H H, H H : H x N H Z y xy H H H H x H Aut H H H,,, x, y N g A Q x, y N g S H K 10

11 SL2,3 2 C Q, S A, B A, B H H H H t 1mod p p t r 1mod p p r H H Tn, F φp g g a b q a τ b a a q N H a N 1 ΦP 1 N H N a 11

12 א א א ROUPS SERIES : : ( ), :, AB ab: a A, b B : A AB ab ab: b B : B AB Ab ab: a A : ( ) H H : gh Hg, g : ( ) : N an; a, N 12

13 : anbn abn, N N N x x N xn : ( ) g g : ( ) : : :, :, : K H K H 13

14 :() : H H : ( ) H H : H : H : ( ) A A N A x ; A A, A x : : ( ) : Z x ; s s ; s : ( ) p p 18 n 0 p 14

15 : ( ) p P p p P p p : ( ) H Aut (Characteristic subgroup) : : : 1 15

16 12 Kurnosenko,NM,On facterisations of finite groups by supersoluble and nilpotent subgroups, problems in algebra,12, Legchekov H V Criterions of supersolubility of some finite factorizable groups, J Algebra and Discrete math 3, Milne J S, roup theory Amer Math Soc,University of Michigan Robinson D A, Course in the theory of roups, Springer Verlag New York Stein Elementary number theory, A computational Approach, Springer Verlag New York Wielandt, H Finite permutation groups,translated from the erman by R Bercov, Academic, New York, Курош АГ Теория груп 3- изд Наука Москва 1967 г 648 с 19 Шеметков ЛА Формации конечных групп изд Наука Москва 1978 г 271 с 78

17 Study about supersoluble groups Abstract The subject of this thesis is the supersoluble groups, the target of this research is to find certain conditions which are applied on any group to become supersoluble The thesis consists of introduction and three section and conclusion and list of references we studied in first section composition, subnormal, normal series and the Commutators, and we mentioned some of examples on which of them We studied in second section the conception of soluble, polycyclic, and nilpotent groups As we studies in second section the intersection between supersoluble groups in the first side and soluble, polycyclic and nilpotent groups in the second ie, we showed that the supersoluble group is soluble group, but the reflection is not true And the polycyclic group is soluble group, but the reflection is not true And so the nilpotent group is soluble group, but the reflection is not true We have reached throw our studies to this research in the third section to the important theorems And we found new classification for supersoluble group which is deferent from last classification As we mentioned the essential theorem which contain three conditions that applied on groups to become supersoluble group Then we have finished the thesis with list of references throw the thesis 79

18 During the preparation of thesis we do the following: 1-Participating in the first scientific conference of mathematics which held in Albaath University in the period between 14-16\10\ we have also translated the following book: Introduction to group theory 3-we have discussed to magazine of Albaath the research classification of supersoluble group 80

Chief factors. Jack Schmidt. University of Kentucky

Chief 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 information

Algebraic Structures II

Algebraic Structures II MAS 305 Algebraic Structures II Notes 9 Autumn 2006 Composition series soluble groups Definition A normal subgroup N of a group G is called a maximal normal subgroup of G if (a) N G; (b) whenever N M G

More information

NILPOTENT NUMBERS JONATHAN PAKIANATHAN AND KRISHNAN SHANKAR

NILPOTENT 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 information

Math 256 (11) - Midterm Test 2

Math 256 (11) - Midterm Test 2 Name: Id #: Math 56 (11) - Midterm Test Spring Quarter 016 Friday May 13, 016-08:30 am to 09:0 am Instructions: Prob. Points Score possible 1 10 3 18 TOTAL 50 Read each problem carefully. Write legibly.

More information

Finite Groups with ss-embedded Subgroups

Finite 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 information

Math-Net.Ru All Russian mathematical portal

Math-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 information

CONSEQUENCES OF THE SYLOW THEOREMS

CONSEQUENCES 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 information

Math 210A: Algebra, Homework 5

Math 210A: Algebra, Homework 5 Math 210A: Algebra, Homework 5 Ian Coley November 5, 2013 Problem 1. Prove that two elements σ and τ in S n are conjugate if and only if type σ = type τ. Suppose first that σ and τ are cycles. Suppose

More information

Algebra: Groups. Group Theory a. Examples of Groups. groups. The inverse of a is simply a, which exists.

Algebra: Groups. Group Theory a. Examples of Groups. groups. The inverse of a is simply a, which exists. Group Theory a Let G be a set and be a binary operation on G. (G, ) is called a group if it satisfies the following. 1. For all a, b G, a b G (closure). 2. For all a, b, c G, a (b c) = (a b) c (associativity).

More information

MATH 433 Applied Algebra Lecture 19: Subgroups (continued). Error-detecting and error-correcting codes.

MATH 433 Applied Algebra Lecture 19: Subgroups (continued). Error-detecting and error-correcting codes. MATH 433 Applied Algebra Lecture 19: Subgroups (continued). Error-detecting and error-correcting codes. Subgroups Definition. A group H is a called a subgroup of a group G if H is a subset of G and the

More information

GROUPS IN WHICH SYLOW SUBGROUPS AND SUBNORMAL SUBGROUPS PERMUTE

GROUPS 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 information

THE KLEIN GROUP AS AN AUTOMORPHISM GROUP WITHOUT FIXED POINT

THE KLEIN GROUP AS AN AUTOMORPHISM GROUP WITHOUT FIXED POINT PACIFIC JOURNAL OF xmathematics Vol. 18, No. 1, 1966 THE KLEIN GROUP AS AN AUTOMORPHISM GROUP WITHOUT FIXED POINT S. F. BAUMAN An automorphism group V acting on a group G is said to be without fixed points

More information

A Note on Groups with Just-Infinite Automorphism Groups

A Note on Groups with Just-Infinite Automorphism Groups Note di Matematica ISSN 1123-2536, e-issn 1590-0932 Note Mat. 32 (2012) no. 2, 135 140. doi:10.1285/i15900932v32n2p135 A Note on Groups with Just-Infinite Automorphism Groups Francesco de Giovanni Dipartimento

More information

Teddy Einstein Math 4320

Teddy Einstein Math 4320 Teddy Einstein Math 4320 HW4 Solutions Problem 1: 2.92 An automorphism of a group G is an isomorphism G G. i. Prove that Aut G is a group under composition. Proof. Let f, g Aut G. Then f g is a bijective

More information

Rank 3 Latin square designs

Rank 3 Latin square designs Rank 3 Latin square designs Alice Devillers Université Libre de Bruxelles Département de Mathématiques - C.P.216 Boulevard du Triomphe B-1050 Brussels, Belgium adevil@ulb.ac.be and J.I. Hall Department

More information

Higher Algebra Lecture Notes

Higher Algebra Lecture Notes Higher Algebra Lecture Notes October 2010 Gerald Höhn Department of Mathematics Kansas State University 138 Cardwell Hall Manhattan, KS 66506-2602 USA gerald@math.ksu.edu This are the notes for my lecture

More information

ABSTRACT ALGEBRA: REVIEW PROBLEMS ON GROUPS AND GALOIS THEORY

ABSTRACT ALGEBRA: REVIEW PROBLEMS ON GROUPS AND GALOIS THEORY ABSTRACT ALGEBRA: REVIEW PROBLEMS ON GROUPS AND GALOIS THEORY John A. Beachy Northern Illinois University 2000 ii J.A.Beachy This is a supplement to Abstract Algebra, Second Edition by John A. Beachy and

More information

MATH 3005 ABSTRACT ALGEBRA I FINAL SOLUTION

MATH 3005 ABSTRACT ALGEBRA I FINAL SOLUTION MATH 3005 ABSTRACT ALGEBRA I FINAL SOLUTION SPRING 2014 - MOON Write your answer neatly and show steps. Any electronic devices including calculators, cell phones are not allowed. (1) Write the definition.

More information

Recognising nilpotent groups

Recognising 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 information

Check Character Systems Using Chevalley Groups

Check 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 information

GROUP ACTIONS RYAN C. SPIELER

GROUP ACTIONS RYAN C. SPIELER GROUP ACTIONS RYAN C. SPIELER Abstract. In this paper, we examine group actions. Groups, the simplest objects in Algebra, are sets with a single operation. We will begin by defining them more carefully

More information

AN EXTENSION OF YAMAMOTO S THEOREM ON THE EIGENVALUES AND SINGULAR VALUES OF A MATRIX

AN EXTENSION OF YAMAMOTO S THEOREM ON THE EIGENVALUES AND SINGULAR VALUES OF A MATRIX Unspecified Journal Volume 00, Number 0, Pages 000 000 S????-????(XX)0000-0 AN EXTENSION OF YAMAMOTO S THEOREM ON THE EIGENVALUES AND SINGULAR VALUES OF A MATRIX TIN-YAU TAM AND HUAJUN HUANG Abstract.

More information

CHAPTER III NORMAL SERIES

CHAPTER 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 information

Quiz 2 Practice Problems

Quiz 2 Practice Problems Quiz 2 Practice Problems Math 332, Spring 2010 Isomorphisms and Automorphisms 1. Let C be the group of complex numbers under the operation of addition, and define a function ϕ: C C by ϕ(a + bi) = a bi.

More information

The Structure of Minimal Non-ST-Groups

The 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 information

This paper has been published in Journal of Algebra, 319(8): (2008).

This 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 information

1 Finite abelian groups

1 Finite abelian groups Last revised: May 16, 2014 A.Miller M542 www.math.wisc.edu/ miller/ Each Problem is due one week from the date it is assigned. Do not hand them in early. Please put them on the desk in front of the room

More information

ON DOUBLY TRANSITIVE GROUPS OF DEGREE n AND ORDER 2(n-l)n

ON DOUBLY TRANSITIVE GROUPS OF DEGREE n AND ORDER 2(n-l)n ON DOUBLY TRANSITIVE GROUPS OF DEGREE n AND ORDER 2(n-l)n NOBORU ITO Dedicated to the memory of Professor TADASI NAKAYAMA Introduction Let % denote the icosahedral group and let > be the normalizer of

More information

Permutation representations and rational irreducibility

Permutation representations and rational irreducibility Permutation representations and rational irreducibility John D. Dixon School of Mathematics and Statistics Carleton University, Ottawa, Canada March 30, 2005 Abstract The natural character π of a finite

More information

The p-quotient Algorithm

The p-quotient Algorithm The p-quotient Algorithm Marvin Krings November 23, 2016 Marvin Krings The p-quotient Algorithm November 23, 2016 1 / 49 Algorithms p-quotient Algorithm Given a finitely generated group Compute certain

More information

Normal Automorphisms of Free Burnside Groups of Period 3

Normal Automorphisms of Free Burnside Groups of Period 3 Armenian Journal of Mathematics Volume 9, Number, 017, 60 67 Normal Automorphisms of Free Burnside Groups of Period 3 V. S. Atabekyan, H. T. Aslanyan and A. E. Grigoryan Abstract. If any normal subgroup

More information

MORE ON THE SYLOW THEOREMS

MORE ON THE SYLOW THEOREMS MORE ON THE SYLOW THEOREMS 1. Introduction Several alternative proofs of the Sylow theorems are collected here. Section 2 has a proof of Sylow I by Sylow, Section 3 has a proof of Sylow I by Frobenius,

More information

Abstract Algebra: Supplementary Lecture Notes

Abstract Algebra: Supplementary Lecture Notes Abstract Algebra: Supplementary Lecture Notes JOHN A. BEACHY Northern Illinois University 1995 Revised, 1999, 2006 ii To accompany Abstract Algebra, Third Edition by John A. Beachy and William D. Blair

More information

Two subgroups and semi-direct products

Two subgroups and semi-direct products Two subgroups and semi-direct products 1 First remarks Throughout, we shall keep the following notation: G is a group, written multiplicatively, and H and K are two subgroups of G. We define the subset

More information

Sylow subgroups of GL(3,q)

Sylow subgroups of GL(3,q) Jack Schmidt We describe the Sylow p-subgroups of GL(n, q) for n 4. These were described in (Carter & Fong, 1964) and (Weir, 1955). 1 Overview The groups GL(n, q) have three types of Sylow p-subgroups:

More information

Kevin James. p-groups, Nilpotent groups and Solvable groups

Kevin 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 information

1. Group Theory Permutations.

1. 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 information

A Simple Classification of Finite Groups of Order p 2 q 2

A Simple Classification of Finite Groups of Order p 2 q 2 Mathematics Interdisciplinary Research (017, xx yy A Simple Classification of Finite Groups of Order p Aziz Seyyed Hadi, Modjtaba Ghorbani and Farzaneh Nowroozi Larki Abstract Suppose G is a group of order

More information

Difference sets and Hadamard matrices

Difference sets and Hadamard matrices Difference sets and Hadamard matrices Padraig Ó Catháin University of Queensland 5 November 2012 Outline 1 Hadamard matrices 2 Symmetric designs 3 Hadamard matrices and difference sets 4 Two-transitivity

More information

Exercises on chapter 1

Exercises on chapter 1 Exercises on chapter 1 1. Let G be a group and H and K be subgroups. Let HK = {hk h H, k K}. (i) Prove that HK is a subgroup of G if and only if HK = KH. (ii) If either H or K is a normal subgroup of G

More information

Modern Algebra I. Circle the correct answer; no explanation is required. Each problem in this section counts 5 points.

Modern Algebra I. Circle the correct answer; no explanation is required. Each problem in this section counts 5 points. 1 2 3 style total Math 415 Please print your name: Answer Key 1 True/false Circle the correct answer; no explanation is required. Each problem in this section counts 5 points. 1. Every group of order 6

More information

Name: Solutions - AI FINAL EXAM

Name: Solutions - AI FINAL EXAM 1 2 3 4 5 6 7 8 9 10 11 12 13 total Name: Solutions - AI FINAL EXAM The first 7 problems will each count 10 points. The best 3 of # 8-13 will count 10 points each. Total is 100 points. A 4th problem from

More information

3.8 Cosets, Normal Subgroups, and Factor Groups

3.8 Cosets, Normal Subgroups, and Factor Groups 3.8 J.A.Beachy 1 3.8 Cosets, Normal Subgroups, and Factor Groups from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 29. Define φ : C R by φ(z) = z, for

More information

The Cyclic Subgroup Separability of Certain HNN Extensions

The Cyclic Subgroup Separability of Certain HNN Extensions BULLETIN of the Malaysian Mathematical Sciences Society http://math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. (2) 29(1) (2006), 111 117 The Cyclic Subgroup Separability of Certain HNN Extensions 1

More information

1 Chapter 6 - Exercise 1.8.cf

1 Chapter 6 - Exercise 1.8.cf 1 CHAPTER 6 - EXERCISE 1.8.CF 1 1 Chapter 6 - Exercise 1.8.cf Determine 1 The Class Equation of the dihedral group D 5. Note first that D 5 = 10 = 5 2. Hence every conjugacy class will have order 1, 2

More information

Algebra SEP Solutions

Algebra SEP Solutions Algebra SEP Solutions 17 July 2017 1. (January 2017 problem 1) For example: (a) G = Z/4Z, N = Z/2Z. More generally, G = Z/p n Z, N = Z/pZ, p any prime number, n 2. Also G = Z, N = nz for any n 2, since

More information

Block-Transitive 4 (v, k, 4) Designs and Suzuki Groups

Block-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 information

C-Characteristically Simple Groups

C-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 information

Homomorphisms. The kernel of the homomorphism ϕ:g G, denoted Ker(ϕ), is the set of elements in G that are mapped to the identity in G.

Homomorphisms. The kernel of the homomorphism ϕ:g G, denoted Ker(ϕ), is the set of elements in G that are mapped to the identity in G. 10. Homomorphisms 1 Homomorphisms Isomorphisms are important in the study of groups because, being bijections, they ensure that the domain and codomain groups are of the same order, and being operation-preserving,

More information

Problem Set Mash 1. a2 b 2 0 c 2. and. a1 a

Problem Set Mash 1. a2 b 2 0 c 2. and. a1 a Problem Set Mash 1 Section 1.2 15. Find a set of generators and relations for Z/nZ. h 1 1 n 0i Z/nZ. Section 1.4 a b 10. Let G 0 c a, b, c 2 R,a6 0,c6 0. a1 b (a) Compute the product of 1 a2 b and 2 0

More information

Rapporto di Ricerca CS G. Busetto, E. Jabara

Rapporto di Ricerca CS G. Busetto, E. Jabara UNIVERSITÀ CA FOSCARI DI VENEZIA Dipartimento di Informatica Technical Report Series in Computer Science Rapporto di Ricerca CS-2005-12 Ottobre 2005 G. Busetto, E. Jabara Some observations on factorized

More information

Ph.D. Qualifying Examination in Algebra Department of Mathematics University of Louisville January 2018

Ph.D. Qualifying Examination in Algebra Department of Mathematics University of Louisville January 2018 Ph.D. Qualifying Examination in Algebra Department of Mathematics University of Louisville January 2018 Do 6 problems with at least 2 in each section. Group theory problems: (1) Suppose G is a group. The

More information

Course 311: Abstract Algebra Academic year

Course 311: Abstract Algebra Academic year Course 311: Abstract Algebra Academic year 2007-08 D. R. Wilkins Copyright c David R. Wilkins 1997 2007 Contents 1 Topics in Group Theory 1 1.1 Groups............................... 1 1.2 Examples of Groups.......................

More information

INTRODUCTION TO THE GROUP THEORY

INTRODUCTION TO THE GROUP THEORY Lecture Notes on Structure of Algebra INTRODUCTION TO THE GROUP THEORY By : Drs. Antonius Cahya Prihandoko, M.App.Sc e-mail: antoniuscp.fkip@unej.ac.id Mathematics Education Study Program Faculty of Teacher

More information

Difference sets and Hadamard matrices

Difference sets and Hadamard matrices Difference sets and Hadamard matrices Padraig Ó Catháin National University of Ireland, Galway 14 March 2012 Outline 1 (Finite) Projective planes 2 Symmetric Designs 3 Difference sets 4 Doubly transitive

More information

PROBLEMS FROM GROUP THEORY

PROBLEMS FROM GROUP THEORY PROBLEMS FROM GROUP THEORY Page 1 of 12 In the problems below, G, H, K, and N generally denote groups. We use p to stand for a positive prime integer. Aut( G ) denotes the group of automorphisms of G.

More information

Graded modules over classical simple Lie algebras

Graded modules over classical simple Lie algebras Graded modules over classical simple Lie algebras Alberto Elduque Universidad de Zaragoza (joint work with Mikhail Kochetov) Graded modules. Main questions Graded Brauer group Brauer invariant Solution

More information

An arithmetic theorem related to groups of bounded nilpotency class

An 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 information

HOMEWORK 3 LOUIS-PHILIPPE THIBAULT

HOMEWORK 3 LOUIS-PHILIPPE THIBAULT HOMEWORK 3 LOUIS-PHILIPPE THIBAULT Problem 1 Let G be a group of order 56. We have that 56 = 2 3 7. Then, using Sylow s theorem, we have that the only possibilities for the number of Sylow-p subgroups

More information

[x, y] =x 1 y 1 xy. Definition 7.1 Let A and B be subgroups of a group G. Define the commutator. [A, B] = [a, b] a A, b B,

[x, y] =x 1 y 1 xy. Definition 7.1 Let A and B be subgroups of a group G. Define the commutator. [A, B] = [a, b] a A, b B, Chapter 7 Nilpotent Groups Recall the commutator is given by [x, y] =x 1 y 1 xy. Definition 7.1 Let A and B be subgroups of a group G. Define the commutator subgroup [A, B] by [A, B] = [a, b] a A, b B,

More information

A Family of One-regular Graphs of Valency 4

A Family of One-regular Graphs of Valency 4 Europ. J. Combinatorics (1997) 18, 59 64 A Family of One-regular Graphs of Valency 4 D RAGAN M ARUSä ICä A graph is said to be one - regular if its automorphism group acts regularly on the set of its arcs.

More information

Converse to Lagrange s Theorem Groups

Converse to Lagrange s Theorem Groups Converse to Lagrange s Theorem Groups Blain A Patterson Youngstown State University May 10, 2013 History In 1771 an Italian mathematician named Joseph Lagrange proved a theorem that put constraints on

More information

Group Actions Definition. Let G be a group, and let X be a set. A left action of G on X is a function θ : G X X satisfying:

Group Actions Definition. Let G be a group, and let X be a set. A left action of G on X is a function θ : G X X satisfying: Group Actions 8-26-202 Definition. Let G be a group, and let X be a set. A left action of G on X is a function θ : G X X satisfying: (a) θ(g,θ(g 2,x)) = θ(g g 2,x) for all g,g 2 G and x X. (b) θ(,x) =

More information

On the characterization of the numbers n such that any group of order n has a given property P

On the characterization of the numbers n such that any group of order n has a given property P On the characterization of the numbers n such that any group of order n has a given property P arxiv:1501.03170v1 [math.gr] 6 Jan 2015 Advisor: Professor Thomas Haines Honors Thesis in Mathematics University

More information

MATHEMATICS COMPREHENSIVE EXAM: IN-CLASS COMPONENT

MATHEMATICS COMPREHENSIVE EXAM: IN-CLASS COMPONENT MATHEMATICS COMPREHENSIVE EXAM: IN-CLASS COMPONENT The following is the list of questions for the oral exam. At the same time, these questions represent all topics for the written exam. The procedure for

More information

The Class Equation X = Gx. x X/G

The 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 information

arxiv: v4 [math.gr] 17 Jun 2015

arxiv: v4 [math.gr] 17 Jun 2015 On finite groups all of whose cubic Cayley graphs are integral arxiv:1409.4939v4 [math.gr] 17 Jun 2015 Xuanlong Ma and Kaishun Wang Sch. Math. Sci. & Lab. Math. Com. Sys., Beijing Normal University, 100875,

More information

arxiv: v1 [math.gr] 6 Nov 2017

arxiv: 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 information

LIE RING METHODS IN THE THEORY OF FINITE NILPOTENT GROUPS

LIE RING METHODS IN THE THEORY OF FINITE NILPOTENT GROUPS 307 LIE RING METHODS IN THE THEORY OF FINITE NILPOTENT GROUPS By GRAHAM HIGMAN 1. Introduction There are, of course, many connections between Group Theory and the theory of Lie rings, and my title is,

More information

arxiv:math/ v1 [math.gr] 15 Jun 1994

arxiv:math/ v1 [math.gr] 15 Jun 1994 DIMACS Series in Discrete Mathematics and Theoretical Computer Science Volume 00, 0000 arxiv:math/9406207v1 [math.gr] 15 Jun 1994 Application of Computational Tools for Finitely Presented Groups GEORGE

More information

MATH EXAMPLES: GROUPS, SUBGROUPS, COSETS

MATH EXAMPLES: GROUPS, SUBGROUPS, COSETS MATH 370 - EXAMPLES: GROUPS, SUBGROUPS, COSETS DR. ZACHARY SCHERR There seemed to be a lot of confusion centering around cosets and subgroups generated by elements. The purpose of this document is to supply

More information

Math 3140 Fall 2012 Assignment #3

Math 3140 Fall 2012 Assignment #3 Math 3140 Fall 2012 Assignment #3 Due Fri., Sept. 21. Remember to cite your sources, including the people you talk to. My solutions will repeatedly use the following proposition from class: Proposition

More information

FREE PRODUCTS AND BRITTON S LEMMA

FREE PRODUCTS AND BRITTON S LEMMA FREE PRODUCTS AND BRITTON S LEMMA Dan Lidral-Porter 1. Free Products I determined that the best jumping off point was to start with free products. Free products are an extension of the notion of free groups.

More information

A Note on Just-Non-X Groups

A Note on Just-Non-X Groups International Journal of Algebra, Vol. 2, 2008, no. 6, 277-290 A Note on Just-Non-X Groups Francesco Russo Department of Mathematics University of Naples, Naples, Italy francesco.russo@dma.unina.it Abstract

More information

THE /^-PROBLEM AND THE STRUCTURE OF tf

THE /^-PROBLEM AND THE STRUCTURE OF tf THE /^-PROBLEM AND THE STRUCTURE OF tf D. R. HUGHES AND J. G. THOMPSON 1* Introduction* Let G be a group, p a prime, and H P (G) the subgroup of G generated by the elements of G which do not have order

More information

Elements with Square Roots in Finite Groups

Elements with Square Roots in Finite Groups Elements with Square Roots in Finite Groups M. S. Lucido, M. R. Pournaki * Abstract In this paper, we study the probability that a randomly chosen element in a finite group has a square root, in particular

More information

Connectivity of Cayley Graphs: A Special Family

Connectivity of Cayley Graphs: A Special Family Connectivity of Cayley Graphs: A Special Family Joy Morris Department of Mathematics and Statistics Trent University Peterborough, Ont. K9J 7B8 January 12, 2004 1 Introduction Taking any finite group G,

More information

Some practice problems for midterm 2

Some practice problems for midterm 2 Some practice problems for midterm 2 Kiumars Kaveh November 14, 2011 Problem: Let Z = {a G ax = xa, x G} be the center of a group G. Prove that Z is a normal subgroup of G. Solution: First we prove Z is

More information

RELATIVE N-TH NON-COMMUTING GRAPHS OF FINITE GROUPS. Communicated by Ali Reza Ashrafi. 1. Introduction

RELATIVE N-TH NON-COMMUTING GRAPHS OF FINITE GROUPS. Communicated by Ali Reza Ashrafi. 1. Introduction Bulletin of the Iranian Mathematical Society Vol. 39 No. 4 (2013), pp 663-674. RELATIVE N-TH NON-COMMUTING GRAPHS OF FINITE GROUPS A. ERFANIAN AND B. TOLUE Communicated by Ali Reza Ashrafi Abstract. Suppose

More information

DIHEDRAL GROUPS OF ORDER 2 m+1. Al Hussein Bin Talal University Maan, JORDAN 2 Department of Mathematics

DIHEDRAL GROUPS OF ORDER 2 m+1. Al Hussein Bin Talal University Maan, JORDAN 2 Department of Mathematics International Journal of Applied Mathematics Volume 26 No. 1 2013, 1-7 ISSN: 1311-1728 (printed version); ISSN: 1314-8060 (on-line version) doi: http://dx.doi.org/10.12732/ijam.v26i1.1 DIHEDRAL GROUPS

More information

Written Homework # 1 Solution

Written Homework # 1 Solution Math 516 Fall 2006 Radford Written Homework # 1 Solution 10/11/06 Remark: Most of the proofs on this problem set are just a few steps from definitions and were intended to be a good warm up for the course.

More information

YVES BENOIST AND HEE OH. dedicated to Professor Atle Selberg with deep appreciation

YVES BENOIST AND HEE OH. dedicated to Professor Atle Selberg with deep appreciation DISCRETE SUBGROUPS OF SL 3 (R) GENERATED BY TRIANGULAR MATRICES YVES BENOIST AND HEE OH dedicated to Professor Atle Selberg with deep appreciation Abstract. Based on the ideas in some recently uncovered

More information

1. Group actions and other topics in group theory

1. Group actions and other topics in group theory 1. Group actions and other topics in group theory October 11, 2014 The main topics considered here are group actions, the Sylow theorems, semi-direct products, nilpotent and solvable groups, and simple

More information

BASIC GROUP THEORY : G G G,

BASIC GROUP THEORY : G G G, BASIC GROUP THEORY 18.904 1. Definitions Definition 1.1. A group (G, ) is a set G with a binary operation : G G G, and a unit e G, possessing the following properties. (1) Unital: for g G, we have g e

More information

Formulary for elliptic divisibility sequences and elliptic nets. Let E be the elliptic curve defined over the rationals with Weierstrass equation

Formulary for elliptic divisibility sequences and elliptic nets. Let E be the elliptic curve defined over the rationals with Weierstrass equation Formulary for elliptic divisibility sequences and elliptic nets KATHERINE E STANGE Abstract Just the formulas No warranty is expressed or implied May cause side effects Not to be taken internally Remove

More information

VARIATIONS ON THE BAER SUZUKI THEOREM. 1. Introduction

VARIATIONS ON THE BAER SUZUKI THEOREM. 1. Introduction VARIATIONS ON THE BAER SUZUKI THEOREM ROBERT GURALNICK AND GUNTER MALLE Dedicated to Bernd Fischer on the occasion of his 75th birthday Abstract. The Baer Suzuki theorem says that if p is a prime, x is

More information

FROBENIUS SUBGROUPS OF FREE PROFINITE PRODUCTS

FROBENIUS SUBGROUPS OF FREE PROFINITE PRODUCTS FROBENIUS SUBGROUPS OF FREE PROFINITE PRODUCTS ROBERT M. GURALNICK AND DAN HARAN Abstract. We solve an open problem of Herfort and Ribes: Profinite Frobenius groups of certain type do occur as closed subgroups

More information

On the structure of some modules over generalized soluble groups

On 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 information

AUTOMORPHISMS OF A -GROUP

AUTOMORPHISMS OF A -GROUP AUTOMORPHISMS OF A -GROUP BY J E Ao Ao T 1 Introduction There hve been number of results on the relationship between the order of finite group G nd the order of its automorphism group A A (G), for example,

More information

RINGS GRADED BY POLYCYCLIC-BY-FINITE WILLIAM CHIN AND DECLAN QUINN. (Communicated by Donald Passman)

RINGS GRADED BY POLYCYCLIC-BY-FINITE WILLIAM CHIN AND DECLAN QUINN. (Communicated by Donald Passman) PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 102, Number 2, February 1988 RINGS GRADED BY POLYCYCLIC-BY-FINITE WILLIAM CHIN AND DECLAN QUINN (Communicated by Donald Passman) GROUPS ABSTRACT.

More information

Section V.9. Radical Extensions

Section V.9. Radical Extensions V.9. Radical Extensions 1 Section V.9. Radical Extensions Note. In this section (and the associated appendix) we resolve the most famous problem from classical algebra using the techniques of modern algebra

More information

SOME RESIDUALLY FINITE GROUPS SATISFYING LAWS

SOME RESIDUALLY FINITE GROUPS SATISFYING LAWS SOME RESIDUALLY FINITE GROUPS SATISFYING LAWS YVES DE CORNULIER AND AVINOAM MANN Abstract. We give an example of a residually-p finitely generated group, that satisfies a non-trivial group law, but is

More information

Homework #11 Solutions

Homework #11 Solutions Homework #11 Solutions p 166, #18 We start by counting the elements in D m and D n, respectively, of order 2. If x D m and x 2 then either x is a flip or x is a rotation of order 2. The subgroup of rotations

More information

Definition 1.2. Let G be an arbitrary group and g an element of G.

Definition 1.2. Let G be an arbitrary group and g an element of G. 1. Group Theory I Section 2.2 We list some consequences of Lagrange s Theorem for exponents and orders of elements which will be used later, especially in 2.6. Definition 1.1. Let G be an arbitrary group

More information

PROFINITE GROUPS WITH RESTRICTED CENTRALIZERS

PROFINITE GROUPS WITH RESTRICTED CENTRALIZERS proceedings of the american mathematical society Volume 122, Number 4, December 1994 PROFINITE GROUPS WITH RESTRICTED CENTRALIZERS ANER SHALEV (Communicated by Ronald M. Solomon) Abstract. Let G be a profinite

More information

1. Let r, s, t, v be the homogeneous relations defined on the set M = {2, 3, 4, 5, 6} by

1. Let r, s, t, v be the homogeneous relations defined on the set M = {2, 3, 4, 5, 6} by Seminar 1 1. Which ones of the usual symbols of addition, subtraction, multiplication and division define an operation (composition law) on the numerical sets N, Z, Q, R, C? 2. Let A = {a 1, a 2, a 3 }.

More information

Cyclic Groups. AgroupG is called cyclic if there is an element x G such that for each a G, a = x n for some n Z. G = { x k : k Z }.

Cyclic Groups. AgroupG is called cyclic if there is an element x G such that for each a G, a = x n for some n Z. G = { x k : k Z }. Cyclic Groups AgroupG is called cyclic if there is an element x G such that for each a G, a = x n for some n Z. In other words, G = { x k : k Z }. G is said to be generated by x, denoted by G = x. x is

More information

Self-complementary circulant graphs

Self-complementary circulant graphs Self-complementary circulant graphs Brian Alspach Joy Morris Department of Mathematics and Statistics Burnaby, British Columbia Canada V5A 1S6 V. Vilfred Department of Mathematics St. Jude s College Thoothoor

More information

ELEMENTARY GROUPS BY HOMER BECHTELL

ELEMENTARY GROUPS BY HOMER BECHTELL ELEMENTARY GROUPS BY HOMER BECHTELL 1. Introduction. The purpose of this paper is to investigate two classes of finite groups, elementary groups and E-groups. The elementary groups have the property that

More information

Engel Groups (a survey) Gunnar Traustason Department of Mathematical Sciences University of Bath

Engel Groups (a survey) Gunnar Traustason Department of Mathematical Sciences University of Bath Engel Groups (a survey) Gunnar Traustason Department of Mathematical Sciences University of Bath Definition. Let G be a group and a G. (a) We say that G is an Engel group if for each pair (x, y) G G there

More information