COMMUTATIVITY DEGREES AND RELATED INVARIANTS OF SOME FINITE NILPOTENT GROUPS FADILA NORMAHIA BINTI ABD MANAF UNIVERSITI TEKNOLOGI MALAYSIA

Transcription

1 COMMUTATIVITY DEGREES AND RELATED INVARIANTS OF SOME FINITE NILPOTENT GROUPS FADILA NORMAHIA BINTI ABD MANAF UNIVERSITI TEKNOLOGI MALAYSIA

2 COMMUTATIVITY DEGREES AND RELATED INVARIANTS OF SOME FINITE NILPOTENT GROUPS FADILA NORMAHIA BINTI ABD MANAF A thesis submitted i fulfilmet of the requiremets for the award of the degree of Doctor of Philosophy (Mathematics) Faculty of Sciece Uiversiti Tekologi Malaysia MARCH 2013

3 iii To My Mom ad Dad To My Beloved Husbad To My Family

4 iv ACKNOWLEDGEMENTS I the ame of Allah, the most gracious ad merciful. He who has give me stregth, courage ad calm i completig my research. First of all, I would love to express my greatest gratitude to my beloved supervisor, Assoc. Prof. Dr. Nor Haiza Sarmi for her guidace, motivatio, patiece ad icredible help throughout the process of doig this thesis util the ed. I am also thakful to both of my co-supervisors, Dr Nor Muhaiiah Mohd Ali ad Prof Dr Ahmad Erfaia for their helps ad for sharig their expertise durig my studies. I wat to thak my co-author, Dr Fracesco Russo from the Departmet of Electrical, Electroics ad Telecommuicatios Egieerig (DIEETCAM), Uiversity of Palermo, Italy for his assistace i completig some parts of this thesis. I am grateful for his direct supervisio while I was at Piedimote Matese, Italy, for a two-week attachmet (15 29 April 2012). Thaks also to Behaz Tolue ad Mohammad Farrokhi from Ferdowsi Uiversity of Mashhad, Ira, for their helps i completig parts of my research. The discussios I had with them were the most ivaluable oes I have ever experieced. My special thaks to my family, especially my parets, for their cotiuous support ad patiece. Last but ot least, thaks to my beloved husbad, Muhammad Hafiz Mohd Hamda ad my dearest fried, Hazzirah Izzati Mat Hassim who have helped me i may ways i the completio of this thesis. Thak you.

5 v ABSTRACT I this research, two-geerator p-groups of ilpotecy class two, which is referred to as G are cosidered. The commutativity degree of a fiite group G, deoted as P G, is defied as the probability that a radom elemet of the group G commutes with aother radom elemet i G. The mai objective of this research is to derive the geeral formula for P G ad its geeralizatios. This research starts by fidig the formula for the umber of cojugacy classes of G. The the commutativity degree of each of these groups is determied by usig the fact that the commutativity degree of a fiite group G is equal to the umber of cojugacy classes of G divided by the order of G. The commutativity degree ca be geeralized to the cocepts of -th commutativity degree, P G, which is defied as the probability that the -th power of a radom elemet commutes with aother radom elemet from the same group. Moreover, P G commutativity degree, ca be exteded to the relative -th P H, G, which is the probability of commutig the -th power of a radom elemet of H with a elemet of G, where H is a subgroup of G. I this research, the explicit formulas for P G ad, P H G are computed. Meawhile, aother geeralizatio of the commutativity degree, which is called commutator degree ad deoted by Pg G, is the probability that the commutator of two elemets i G is equal to a elemet g i G. I this research, a effective character-free method is used for fidig the exact formula for Pg G. ^ exterior degree of the wreath product of A ad B, P A B are two fiite abelia groups., Fially, the is foud where A ad B

6 vi ABSTRAK Dalam peyelidika ii, kumpula-p berpejaa-dua dega kelas ilpote dua, dirujuk sebagai G, dipertimbagka. Darjah kekalisa tukar tertib bagi suatu kumpula terhigga G, ditadaka sebagai P G, ditakrifka sebagai kebaragkalia bahawa satu usur dalam G yag dipilih secara rawak adalah kalis tukar tertib dega usur lai yag dipilih secara rawak dalam G. Objektif utama kajia ii adalah utuk medapatka rumus umum bagi PG P G. da pegitlaka bagi Kajia ii dimulaka dega mecari rumus utuk bilaga kelas kekojugata bagi G. Kemudia darjah kekalisa tukar tertib bagi kumpulakumpula ii ditetuka dega megguaka fakta bahawa darjah kekalisa tukar tertib bagi suatu kumpula terhigga G adalah sama dega bilaga kelas kekojugata dibahagi dega perigkat kumpula. Darjah kekalisa tukar tertib boleh diitlakka kepada kosep darjah kekalisa tukar tertib kali ke-, P G, yag ditakrifka sebagai kebaragkalia bahawa kuasa ke- bagi suatu usur yag dipilih secara rawak berkalis tukar tertib dega usur yag lai daripada kumpula yag sama. Selai itu, P G tertib secara relatif, P H, G boleh dilajutka kepada kosep darjah kekalisa tukar yag ditakrifka sebagai kebaragkalia berkalisa tukar tertib kuasa ke- bagi suatu usur dalam H yag dipilih secara rawak dega suatu usur dalam G, dega H adalah sub-kumpula bagi G. Dalam kajia ii, rumus eksplisit utuk P G da P H, G dikira. Semetara itu, pegitlaka yag lai bagi darjah kekalisa tukar tertib, yag diamai darjah pegalis tukar tertib da dilambagi Pg G, merupaka kebaragkalia bahawa peukar tertib bagi dua usur dalam kumpula G adalah sama dega usur g yag diberika dalam kumpula tersebut. Dalam peyelidika ii, satu kaedah bebas-aksara yag berkesa telah diguaka utuk mecari rumus tepat bagi Pg G. ^ peluara bagi hasil darab kaluga A da B, P A B kumpula abela terhigga A da B., Akhir sekali, darjah diperoleh bagi dua

7 vii TABLE OF CONTENTS CHAPTER TITLE PAGE DECLARATION DEDICATION ACKNOWLEDGEMENTS ABSTRACT ABSTRAK TABLE OF CONTENTS LIST OF SYMBOLS ii iii iv v vi vii x 1 INTRODUCTION Itroductio Research Backgroud Commutativity Degree ad Its Geeralizatios Motivatio Problem Statemets Research Objectives Scope of the Study Sigificace of Fidigs Research Methodology Groups, Algorithms ad Programmig (GAP) Thesis Outlie 7

8 viii 2 LITERATURE REVIEW Itroductio The Commutativity Degree of a Fiite Group The -th ad Relative -th Commutativity Degree of a 12 Fiite Group 2. 4 The Commutator Degree of a Fiite Group The Exterior Degree of a Fiite Group The Classificatio of Two-Geerator p-groups of 16 Nilpotecy Class Two 2. 7 Coclusio 19 3 THE COMMUTATIVITY DEGREE OF FINITE NILPOTENT GROUPS OF CLASS TWO Itroductio Prelimiary Results The Commutativity Degree of Two-Geerator p- Groups of Nilpotecy Class Two (p a odd prime) 3.4 The Commutativity Degree of Two-Geerator p- Groups of Nilpotecy Class Two ( p 2 ) 3.5 Coclusio THE -TH AND RELATIVE -TH COMMUTATIVITY DEGREE OF FINITE NILPOTENT GROUPS OF CLASS TWO Itroductio Prelimiary Results The -th Commutativity Degree of Two-Geerator p- 41 Groups of Nilpotecy Class Two 4.4 The Relative -th Commutativity Degree of Two- 45 Geerator p-groups of Nilpotecy Class Two 4.5 Coclusio 48

9 ix 5 COMMUTATOR DEGREE OF FINITE NILPOTENT 49 GROUPS OF CLASS TWO 5.1 Itroductio Prelimiary Results The Commutator Degree of Two-Geerator p-groups 53 of Nilpotecy Class Two 5.4 Coclusio 57 6 THE EXTERIOR DEGREE OF FINITE NILPOTENT 58 GROUPS OF CLASS TWO 6.1 Itroductio Prelimiary Results The Exterior Degree of the Wreath Product of Two 63 Fiite Abelia Groups 6.4 Coclusio 66 7 CONCLUSION Summary of the Research Suggestios for Future Research 68 REFERENCES 70 Appedix A 75 Appedix B 76

10 x LIST OF SYMBOLS PG - Commutativity degree of a group G. P G - -th commutativity degree of a group G. P H, G - Relative -th commutativity degree of a group G. g P G - Commutator degree of a group G. P G - Exterior degree of a group G. G G - Exterior square of a group G. 1 - The idetity of a group G. 1 - The idetity of exterior square of a group G. G G - Noabelia tesor square of a group G. M G - Schur multiplier. k G - The umber of cojugacy classes of a group G. i i G - The group geerated by all elemets x G such that x p 1, C Z G x G for every iteger i 0. - Exterior cetralizer. - Exterior ceter of a group G. A B - The wreath product of two fiite abelia groups A ad B. A# A - The factor group of A A over the subgroup geerated by the elemets of the form a b b a. Iv A - A elemet of order two i ay group A. G H - The factor group of H i G.

11 CHAPTER 1 INTRODUCTION 1.1 Itroductio The probability that two elemets of a group commute is called the commutativity degree of the group G, which is deoted by PG. This otio was first itroduced by Miller [1] i 1944 ad has bee geeralized i a umber of ways. Mohd Ali ad Sarmi [2] i 2006 exteded the defiitio of commutativity degree ad defied a ew geeralizatio of this degree which is called the -th commutativity degree, P G. This cocept ca be writte as the probability that the -th power of a radom elemet commutes with aother radom elemet from the same group, G. A few years later, Erfaia et al. [3] gave the relative case of - th commutativity degree. They idetify the probability that the -th power of a radom elemet of a subgroup H of G commutes with aother radom elemet of G, deoted as P H G,. The commutativity degree ca also be defied as the probability that the commutator of two elemets i a group, G, is equal to idetity. Therefore, usig this defiitio, Pouraki ad Sobhai [4] provide a ew geeralizatio of commutativity degree, that is, the probability that the commutator of two elemets i a group, G, is equal to a give elemet. I this research, this probability is defied as the commutator degree.

12 2 The exterior square of a group G, deoted as G^ G GG G G^ G, is defied as where G G is the oabelia tesor square of G ad ( G) is the subgroup of G geerated by x x for x G. Recetly, Niroomad ad Rezaei [5] gave some relatios betwee the cocept of exterior square ad commutativity degree by defiig the exterior degree of a group G, P ( G), as the probability that two elemets g ad g ' i G satisfy gg' 1, where 1 is the idetity of G G. I this research, the commutativity degree, -th commutativity degree, relative -th commutativity degree ad the commutator degree for two-geerator p- groups of ilpotecy class two are determied. Moreover, a upper ad lower boud of the wreath product of two abelia groups are foud. 1.2 Research Backgroud A group is called a abelia group if every pair of its elemets commutes. This meas that for a group G, ab ba, for all a, b G. However, ot all groups are abelia, the oes that are ot are called o-abelia groups. Ca oe measure i a certai sese how commutative a o-commutative group be? Commutativity Degree ad Its Geeralizatios All groups cosidered i this research are assumed to be fiite. The commutativity degree of a group G, which is deoted by P(G), is the probability that two elemets of the group G, chose radomly with replacemet, commute. This ca be writte as, Number of ordered pairs x, y GG such that xy yx PG. Total umber of ordered pairs x, y GG I other words, the commutativity degree is a kid of measure for the abeliaess of a group.

13 3 As metioed before, there are other geeralizatios of the probability that two elemets commute. Oe of them is the probability that the -th power of a radom elemet commutes with aother radom elemet from the same group, deoted by P (G). The defiitio of P (G) ca be writte as the followig ratio: P G Number of ordered pairs x, y GG such that x y yx Total umber of ordered pairs x, y GG I order to fid P (G), firstly, the power of each elemet is eeded to be gradually raised util the power is achieved. However, if the first copy of G is substituted with a subgroup H of G, the this probability is called a relative -th commutativity degree ad is deoted as P (H,G). That is, P (H,G) is the probability that the -th power of a radom elemet of H commutes with a elemet of G ad ca be writte as: P h, g H G : h, g 1 H, G. H G. The commutativity degree ca be cosidered as a special case of a more geeral probability P g (G), which is defied for each commutator subgroup of G, deoted by P g G, :, x y GG x y g G 2 g G' where G ' is the that is the probability that the commutator of two radomly chose elemets of G is equal to the give elemet g. It is easy to see that P g (G) = P(G) if g 1, the idetity elemet of G., Recetly, the cocept of commutativity degree has bee coected with the exterior square of a group. This coectio is called the exterior degree of a group G, deoted as P ( G), ad is defied as the probability for two elemets x ad y i G to satisfy xy1, where 1 is the idetity of G G. I mathematical symbols, this otio ca be writte as x, y GG x y 1 P ( G). 2 G

14 Motivatio This research has its foudatio with the classificatio of two-geerator p- groups of ilpotecy class two. Baco ad Kappe [6] ad Kappe et al. [7] classified these groups for p is a odd prime ad p 2, respectively. The commutativity degree of a group G ca be computed by dividig the umber of cojugacy classes by the order of the group. The umber of cojugacy classes of two-geerator p- groups of ilpotecy class two where p is a odd prime has bee computed by Ahmad i [8]. Meawhile, Ilagova ad Sarmi [9 12] computed the same for the case p 2. Sice the commutativity degrees for two-geerator p-groups of ilpotecy class two have ot bee foud yet, this research computes the exact formula for these degrees usig the results by Ahmad [8]. The this research also cosiders the other geeralizatios of commutativity degrees which are the -th commutativity degrees, relative -th commutativity degrees ad the commutator degrees. Motivated by a paper writte by Ereveko ad Sury [13] i 2008, this research also foud the exterior degree of the wreath product of two abelia groups. 1.3 Problem Statemets Studets who study probability ad algebra might well ask the followig questios: (i) What is the probability that two elemets of a group commute? (ii) Ca oe measure how commutative a o-commutative group ca be? This research is motivated by those questios. Hece, i this research, the followig questios will be addressed ad aswered. (i) What is the probability that two elemets of a group, chose radomly with replacemet commute? (ii) What is the probability that the -th power of a radom elemet of a group commutes with aother radom elemet from the same group?

15 5 (iii) (iv) What is the probability that the -th power of a radom elemet of a subgroup of a group commutes with a elemet of the group? What is the probability that the commutator of two elemets i a group is equal to a particular elemet of the group? (v) What is the probability that two elemets g ad g ' i a group G satisfy gg' 1, where 1 is the idetity elemet of G G? 1.4 Research Objectives The objectives of this research are: (i) to determie the commutativity degree of two-geerator p-groups of ilpotecy class two ad give the geeralizatio formula. (ii) to compute the -th ad relative -th commutativity degree for twogeerator p-groups of ilpotecy class two ad give the geeralizatio formula. (iii) to fid the commutator degree of a ilpotet p-group of class two. (iv) to obtai the exterior degree of the wreath product of two abelia groups. 1.5 Scope of the Study This thesis focuses o the two-geerator p-groups of ilpotecy class two for characterizig the commutativity degree, -th commutativity degree, relative -th commutativity degree ad the commutator degree. Meawhile, i computig the exterior degree of a group, the wreath product of two abelia groups is cosidered.

16 6 1.6 Sigificace of Fidigs The aim of this research is to preset ew results i group theory i the forms of theorems. Usig these theorems, the commutativity degree, -th commutativity degree, relative -th commutativity degree, commutator degree ad the exterior degree are determied. The methods applied ad the results obtaied ca be used for computig the commutativity degree, -th commutativity degree, relative -th commutativity degree, commutator degree ad the exterior degree for other groups. Besides, the result of the commutativity degree ca be trasferred to a ocommutig graph where this kid of graph ca be used to characterize the group theory properties of a group. For example, if a o-commutig graph of a group G is isomorphic to the o-commutig graph of the alteratig group A where 4, the G A. Research papers have bee accepted ad will be set to be published i idexed local/iteratioal jourals. Research papers also have bee ad will be preseted i local ad iteratioal semiars/cofereces (see Appedix B). 1.7 Research Methodology I the first step, by rewritig the results of [8 12], the exact formula for the umber of cojugacy classes of two geerator p-groups of ilpotecy class two for ay p are foud. The the commutativity degree of these groups is determied by usig the equatio which was itroduced by Gustafso i [14]. I the secod step, the -th commutativity degree ad the relative -th commutativity degree are computed by usig the defiitio of these degrees. The third step is focused o the commutator degree of two-geerator p-groups of ilpotecy class two ad ilpotet p-groups with several coditios. By applyig these results, some theorems are developed. Groups, Algorithms ad Programmig (GAP) software has bee used i some calculatios i Chapter 5. Lastly, bouds for the exterior degree of the wreath product of two abelia groups are foud by computig the Schur multiplier ad cosiderig the bouds of exterior degree of a fiite group.

17 7 1.8 Groups, Algorithms ad Programmig (GAP) Groups, Algorithms ad Programmig (GAP) software is a system which provides a programmig laguage, a library of thousads of fuctios implemetig algebraic algorithms writte i the GAP laguage as well as large data libraries of algebraic objects. This software is used i research ad teachig for studyig groups ad their represetatios, rigs, vector spaces, algebras, combiatorial structures ad some others. GAP also has may built i fuctios, operatios, ad algebraic structures. Thus GAP ca be used to quickly provide umerous examples with more complexity tha could be doe by had. I this research, GAP is used to calculate some of the commutator degrees. 1.9 Thesis Outlie This thesis is divided ito seve chapters which icludes the itroductio, literature review, the commutativity degree of two-geerator p-groups of ilpotecy class two, the -th ad relative -th commutativity degree of two-geerator p-groups of ilpotecy class two, the commutator degree of ilpotet p-groups of class two, the exterior degree of the wreath product of two abelia groups ad coclusio. I the first chapter, the itroductio to the whole thesis is give icludig the research backgroud, problem statemet, research objectives, scope of the study, sigificace of fidigs ad research methodology. This chapter also icludes the cocepts of the commutativity degree, -th commutativity degree, relative -th commutativity degree, the commutator degree ad the exterior degree. Chapter 2 presets the literature review of this research. Various works by differet researchers regardig the commutativity degree ad its geeralizatios are stated. Commutativity degree has bee discovered for 68 years sice 1944 where

18 8 Miller [1] was the first to itroduce the cocept of commutativity degree. The classificatios of two-geerator p-groups of ilpotecy class two for ay prime p are also give i this chapter. Chapter 3 shows a geeral formula for fidig the cojugacy classes of ilpotet p-groups of class two. Besides, this chapter gives the determiatio of commutativity degree of two-geerator p-groups of ilpotecy class two for ay prime p. I additio, some cocepts ad basic results o the commutativity degree are also preseted i this chapter. Next, i Chapter 4, some prelimiary results for -th ad relative -th commutativity degree are icluded. The purpose of this chapter is to compute the explicit formula for P G ad, P H G where G is a two-geerator p-group of ilpotecy class two ad H is a subgroup of G. Furthermore, this chapter shows that if there are two pairs of relative isocliic groups, the they will have equal relative - th commutativity degree, P H G,. I Chapter 5, the precise formulas of the commutator degree for twogeerator p-groups of ilpotecy class two are give. Some computatios of the results of this degree have bee obtaied by usig GAP software. Aother result i this chapter is to determie the commutator degree for special cases of two-geerator p-groups of ilpotecy class two. Chapter 6 focuses o the computatio of the exterior degree of a group which is the wreath product of two abelia groups. Some basic cocepts ad results of exterior square, exterior degree ad the wreath product of two abelia groups are also icluded i this chapter. Fially, the last chapter presets the summary ad coclusio of this research. Some suggestios for future research o the commutativity degree, the -th commutativity degree, the relative -th commutativity degree, the commutator degree ad the exterior degree are also give i this chapter.

19 REFERENCES 1. Miller. G. A. Relative umber of o-ivariat operators i a group. Proc. Nat. Acad. Sci. USA, (2): Mohd Ali. N. M. ad Sarmi. N. H. O some problems i group theory of probabilistic ature. Techical Report of Departmet of Mathematics, Uiversiti Tekologi Malaysia, : Erfaia. A., Tolue. B. ad Sarmi. N. H. Some cosideratios o the -th commutativity degrees of fiite groups. Ars Combiatorial, J I Press. 4. Pouraki. M. R. ad Sobhai. R. Probability that the commutator of two group elemets is equal to a give elemet. J. Pure Appl. Algebra, : Niroomad. P. ad Rezaei. R. O the exterior degree of fiite groups. Comm. Algebra, (1): Baco. M. R. ad Kappe. L-C. The oabelia tesor square of a 2-geerator p-group of class 2. Arch. Math. (Bases), : Kappe. L-C., Sarmi. N. H. ad Visscher. M. P. 2-geerator 2-groups of class 2 ad their oabelia tesor squares. Glasgow Math. J., :

20 71 8. Ahmad. A. The exact umber of cojugacy classes for 2-geerator p-groups of ilpotecy class 2. Ph.D Thesis. Uiversiti Tekologi Malaysia; Ilagova. S. ad Sarmi. N. H. Computig the umber of cojugacy classes of 2-geerator 2-groups of class two. Techical Report of Departmet of Mathematics, Uiversiti Tekologi Malaysia, : Ilagova. S. ad Sarmi. N. H. Computig the umber of cojugacy classes usig Groups, Algorithms ad Programmig (GAP) of some 2-geerator 2- groups of class two. Techical Report of Departmet of Mathematics, Uiversiti Tekologi Malaysia, : Ilagova. S. ad Sarmi. N. H. O the cojugacy classes of split metacyclic 2-groups of class two. Techical Report of Departmet of Mathematics, Uiversiti Tekologi Malaysia, : Ilagova. S. ad Sarmi. N. H. The exact umber of cojugacy classes of some 2-groups of class two. Techical Report of Departmet of Mathematics, Uiversiti Tekologi Malaysia, : Eroveko. I. ad Sury. B. Commutativity degrees of wreath products of fiite abelia groups, Bull. Aust. Math. Soc, : Gustafso. W. H. What is the probability that two group elemets commute? Amer. Math. Mothly, : Erdös. P. ad Tura. P. O some problems of statistical group theory. Acta Math.Acad. of Sci. Hug, : Rusi. D. J. What is the probability that two elemets of a fiite group commute? Pacific Joural of Mathematics, :

21 Sherma. G. The lower boud for the umber of cojugacy classes i fiite ilpotet group. Pacific Joural of Mathematics, : MacHale. D. How commutative ca a o-commutative group be? The Mathematical Gazette, : Lescot. P. Isocliism classes ad commutativity degree of fiite groups. J. Algebra, : Hall. P. The classificatio of prime-power groups, J. Reie Agew Math, : Lescot. P. Cetral extesios ad commutativity degree. Comm. Algebra, (10): Barry. F., MacHale. D. ad Shé. Á. Ní. Some supersolvability coditios for fiite groups. Math. Proc. Royal Irish Acad, A(2): Guralick. R. M. ad Robiso. G. R. O the commutig probability i fiite groups. J. Algebra, (2): Group, T. G. GAP - Groups, Algorithms, ad Programmig, Versio 4.4; Erfaia. A., Rezaei. R. ad Lescot. P. O the relative commutativity degree of a subgroup of a fiite group. Commuicatios i Algebra, : Das. A. K. ad Nath. R. K. O the geeralized relative commutative degree of a fiite group. It. Electr. J. Algebra, :

22 Alghamdi. A. M. A. ad Russo. F. G. A geeralizatio of the probability that the commutator of two group elemets is equal to a give elemet. e-prit, Corell Uiversity, 2010, arxiv: v2 [math.gr]. 28. Brow. R. ad Loday. J. Excisio homotopique e basse dimesio. C. R. Acad. Sci. Ser. I Math. Paris : Brow. R., Johso. D. L. ad Robertso. E. F. Some computatioal of oabelia tesor products of groups. Joural of Algebra, : Niroomad. P. ad Rezaei. R. O the exterior degree of a pair of fiite groups. e-prit, Corell Uiversity, 2011, arxiv: v2 [math.gr]. 31. Rezaei. R., Niroomad. P. ad Erfaia. A. O the multiple exterior degree of fiite groups.. e-prit, Corell Uiversity, 2011, arxiv: v2 [math.gr]. 32. Magidi. A. Capable two-geerator two-groups of class two. Comm. Algebra, (6): Liebeck. M. Graphs whose full automorphism group is a symmetric group, J. Austral. Math. Soc. (Series A), : Xu. C. The commutators of the alteratig group. Sci. Siica, : George McCarty. Topology: a itroductio wit applicatio to topological groups. New York: McGraw Hill, 1967.

23 Beuerle, J. R. ad Kappe, L. C. Ifiite metacyclic groups ad their oabelia tesor squares. Proc. Ediburgh Math. Soc. (2), (3): Baco. M. R. ad Kappe. L-C. O capable p-group of ilpotecy class two. Illiois Joural of Math, : Karpilovsky, G. The Schur Multipler. Oxford: Claredo Press Huppert, B. Edliche Gruppe. Berli: Spriger Verlag Hage. J. ad Harju. T. O ivolutios arisig from graphs, Algorithmic Bioprocesses Spriger Series: Natural Computig Series, Sury. B. Wreath products, Sylow s theorem ad Fermat s little theorem. Europea J. Pure Appl. Math, :