Products of finite groups

347 Pages·2010·1.66 MB·English

by  Mohamed Asaad, Adolfo Ballester-Bolinches, Ramón Esteban-Romero

De Gruyter Expositions in Mathematics 53
Editors
Victor P.Maslov, Academy of Sciences, Moscow
Walter D.Neumann, Columbia University, New York
Raymond O.Wells, Jr., International University, Bremen
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24 EvolutionEquationsandLagrangianCoordinates,A.M.Meirmanov,V.V.Pukhnachov,
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25 CharacterTheoryofFiniteGroups,B.Huppert
26 PositivityinLieTheory:OpenProblems,J.Hilgert,J.D.Lawson,K.-H.Neeb,E.B.Vinberg
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27 AlgebraintheStone-CˇechCompactification,N.Hindman,D.Strauss
28 HolomorphyandConvexityinLieTheory,K.-H.Neeb
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39 Trigonometric Sums in Number Theory and Analysis, G. I. Arkhipov, V. N. Chubarikov,
A.A.Karatsuba
40 EmbeddingProblemsinSymplecticGeometry,F.Schlenk
41 ApproximationsandEndomorphismAlgebrasofModules,R.Göbel,J.Trlifaj
42 SimpleLieAlgebrasoverFieldsofPositiveCharacteristic.II.ClassifyingtheAbsoluteToral
RankTwoCase,H.Strade
43 ModulesoverDiscreteValuationDomains,P.A.Krylov,A.A.Tuganbaev
44 Quasi-Stationary Phenomena in Nonlinearly Perturbed Stochastic Systems, M. Gyllenberg,
D.S.Silvestrov
45 DistributionTheoryofAlgebraicNumbers,P.-C.Hu,C.-C.Yang
46 GroupsofPrimePowerOrder.Volume1,Y.Berkovich
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48 StochasticDynamicsandBoltzmannHierarchy,D.Ya.Petrina
49 AppliedAlgebraicDynamics,V.Anashin,A.Khrennikov
50 UltrafiltersandTopologiesonGroups,Y.G.Zelenyuk
51 StateObserversforLinearSystemswithUncertainty,S.K.Korovin,V.V.Fomichev
52 StabilityAnalysisofImpulsiveFunctionalDifferentialEquations,I.Stamova
Products of Finite Groups
by
Adolfo Ballester-Bolinches
Ramo´n Esteban-Romero
Mohamed Asaad
De Gruyter
MathematicsSubjectClassification2010
Primary20D40,20D10,20D35;
Secondary20D30,20E15,20F16,20F17
ISBN 978-3-11-020417-9
e-ISBN 978-3-11-022061-2
ISSN 0938-6572
LibraryofCongressCataloging-in-PublicationData
Asaad,Mohamed.
Productsoffinitegroups/byMohamedAsaad,AdolfoBallester-
Bolinches,Ramo´nEsteban-Romero.
p.cm.(cid:2)(DeGruyterexpositionsinmathematics;53)
Includesbibliographicalreferencesandindex.
ISBN978-3-11-020417-9(alk.paper)
1.Finitegroups. I.Ballester-Bolinches,Adolfo. II.Esteban-
Romero,Ramo´n. III.Title.
QA177.A78 2010
5121.23(cid:2)dc22
2010031029
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Preface
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The central theme of this book is the structural study of a finite group which can be
factorisedasaproductoffinitelymanypairwisepermutablesubgroups. Theoriginof
the theory may be traced back to the year 1903, when Burnside published his well-
knownpa-lemma;hediscoveredthatafinitegroupcannotbesimpleifitfactorisesas
theproductofaSylowsubgroupandthecentraliserofanon-trivialelement. Oneyear
later, heusedthisresulttoprovehiscelebratedpaqq-theoremaboutthesolubilityof
finitegroupswhoseorderisdivisiblebyatmosttwoprimes,whichisalsoatheorem
about factorised groups. Burnside’s early work was followed by Hall in the decade
1928–1937 in a great sequence of papers which established the basic theory of finite
solublegroups: thisworkdeterminedthedirectionofresearchinfinitesolublegroups
formanyyears,andmayhavewellprovidedthemotivationforanewfieldofresearch
in the theory of factorised groups; he discovered that a finite group is soluble if and
only if it is the product of pairwise permutable Sylow subgroups. But it was a short
paper of Itô in 1955 which set the theory of products in motion. He shows that any
(not necessarily finite) group is metabelian whenever it is the product of two abelian
subgroups. AfterappearanceofItô’stheorem,andmotivatedbytheresultsofBurnside
and Hall, attention is shifted to finite groups that are product of nilpotent subgroups,
the conjecture being that such groups are soluble. In 1958, Wielandt confirms this
conjectureinthecoprimecaseandKegelinthegeneralonein1962. Theoutcomeof
this research is known nowadays as the theorem of Kegel and Wielandt: every finite
group which is factorised as a product of pairwise permutable nilpotent subgroups
is soluble. However it is not known whether its derived length can be bounded in
terms of the nilpotency classes of the factors. Nor it is known whether the Kegel
viii Preface
and Wielandt theorem can be extended to infinite groups. In the much more special
case when the factors are normal and nilpotent, then the product is nilpotent. This is
awell-knownresultofFitting. However,theproductofpairwisenormalsupersoluble
subgroups need not be supersoluble even in the finite case. A natural path of inquiry
opens up when one asks how the structure of the factors affects the structure of the
factorisedgroupwhentheyareconnectedbycertainpermutabilityconditions: totally
andmutuallypermutableproductsnaturallyemerge. ItwasShunkovin1964whofirst
studiedpairwisetotallypermutablesubgroupsofprimepowerorder. Shunkov’searly
workwasfollowedin1989byaseminalpaperofAsaadandShaalanwhichestablished
thebasictheoryoftotallyandmutuallypermutableproducts. Sincethattimetherewas
graduallyemergingabodyoftechniquesandaseriesofgeneralmethodsforstudying
thesekindofproducts. Althoughtheentirefieldispresentlyinastateoffermentand
fluidity,adegreeofstabilityappearstobesettlingovercertainaspectsofthesubject.
Apart from the relevant monographs of N. S. Chernikov [95] (in Russian) of 1987
andtheonebyAmberg,Franciosi,anddeGiovanni[9]of1992,whichcontainmany
beautiful results about products of groups, especially in the infinite case, none of the
existing books on group theory describes more recent developments on products of
finite groups. The present volume has therefore the modest object of presenting in
a unified form a comprehensible account of more than twenty years of work on this
subject, focusing the attention on products of finite nilpotent groups and products of
mutually and totally permutable finite groups. There are some omissions that have
been caused rather by the necessity of keeping the project within reasonable limits.
Thus, for example, the systematic treatment of the finite products of connected sub-
groups with respect to a class of groups, apart from a brief outline without proofs in
Chapter5,hasbeenomitted. Thisisanareathathasbeenthesubjectrecentlyofgreat
dealofinvestigation. Wehavenotdealtwithfactorisationsoffinitesimplegroupsnor
trifactorised groupseither. Nevertheless, wehopethatthisbookwillserveasabasic
referenceinthesubjectarea,asatextforpostgraduatestudies,andalsoasasourceof
researchideas.
Allofourworkisconcernedwithfinitegroups. Thereforeourunspokenruleisthat
all groups are finite. Chapter 1 is foundational material, and it is written on the as-
sumptionthatthereaderhasabasicknowledgeingrouptheory,representationtheory,
and theory of classes of groups, such as which may be obtained from reading Chap-
ters A, B, I, and II of the book of Doerk and Hawkes [119], and Chapters 2 and 4 of
the book of Ballester-Bolinches and Ezquerro [47]. In selecting or rejecting material
for this chapter, we have held steadfast to a single-minded purpose: to present only
those results deemed essential for applications in the rest of the book. In Chapter 2
classes of groups defined by means of certain permutability conditions which have
beenprovedusefulinquestionsinvolvingproductsarediscussed. Amajorpartofthe
chapterisdevotedtocharacterisation theorems. Chapter3providesacomprehensive
lookatproductsoftwonilpotentgroups,focusingtheattentiononproductsofabelian
groups,structuretheorems,andHall–Higmantyperesults. Thestudyofthesubgroup
Preface ix
structureofpairwisemutuallyandtotallypermutableproductsisthemainaimofthe
Chapter 4. The last chapter of the book is dedicated to the study of the behaviour of
suchproductswithrespecttoclassesofgroups.
WewouldliketoacknowledgeourgratitudetoJamesC.Beidleman,JonasBochtler,
ÁngelCarocca,NikolayS.Chernikov,JohnCossey,LuisM.Ezquerro,AndrewFrans-
man, Michiel Hazewinkel, Hermann Heineken, Lev S. Kazarin, Leonid A. Kurda-
chenko, Thomas Masiwa, Victor D. Mazurov, María Carmen Pedraza-Aguilera, and
Alexander N. Skiba. We wish to express our thanks to Robert Plato and Simon Al-
broscheit for their continuous support from Walter de Gruyter during the production
ofthisbook. Finally,wethanktheMinisteriodeEducaciónyCiencia(SpanishGov-
ernment) and FEDER (European Union) for their financial support via the research
grantMTM2007-68010-C03-02.
AdolfoBallester-Bolinches,RamónEsteban-Romero,andMohamedAsaad
Torres-Torres,València,andCairo,August2010