Groups of Prime Power Order: Volume 5

434 Pages·2016·2.511 MB·English

by  Yakov G. Berkovich, Zvonimir Janko

YakovBerkovichandZvonimirJanko
GroupsofPrimePowerOrder
De Gruyter Expositions in
Mathematics
|
Volume 62
Yakov Berkovich and Zvonimir Janko
Groups of Prime
Power Order
|
Volume 5
Edited by
Victor P.Maslov, Moscow, Russia
Walter D.Neumann, New York, USA
Markus J.Pflaum, Boulder, USA
Dierk Schleicher, Bremen, Germany
Raymond O.Wells, Bremen, Germany
MathematicsSubjectClassification2010
20-02,20D15,20E07
Authors
Prof.YakovG.Berkovich
18251Afula
Israel
[email protected]
Prof.Dr.ZvonimirJanko
Ruprecht-Karls-UniversitätHeidelberg
MathematischesInstitut
ImNeuenheimerFeld288
69120Heidelberg
Germany
[email protected]
ISBN978-3-11-029534-4
e-ISBN(PDF)978-3-11-029535-1
e-ISBN(EPUB)978-3-11-038904-3
Set-ISBN978-3-11-029536-8
ISSN0938-6572
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Contents
Listofdefinitionsandnotations|XIII
Preface|XIX
§190 Onp-groupscontainingasubgroupofmaximalclassandindexp|1
§191 p-groupsGallofwhosenonnormalsubgroupscontainG󸀠initsnormal
closure|4
§192 p-groupswithallsubgroupsisomorphictoquotientgroups|7
§193 Classificationofp-groupsallofwhosepropersubgroupsare
s-self-dual|15
§194 p-groupsallofwhosemaximalsubgroups,exceptone,are
s-self-dual|30
§195 Nonabelianp-groupsallofwhosesubgroupsareq-self-dual|33
§196 Ap-groupwithabsolutelyregularnormalizerofsomesubgroup|40
§197 Minimalnon-q-self-dual2-groups|43
§198 Nonmetacyclicp-groupswithmetacycliccentralizerofanelementof
orderp|52
§199 p-groupswithminimalnonabelianclosuresofallnonnormalabelian
subgroups |56
§200 Thenonexistenceofp-groupsGallofwhoseminimalnonabelian
subgroupsintersectZ(G)trivially|61
§201 Subgroupsoforderppandexponentpinp-groupswithanirregular
subgroupofmaximalclassandindex>p|64
§202 p-groupsallofwhoseA2-subgroupsaremetacyclic|67
§203 Nonabelianp-groupsGinwhichthecenterofeachnonabeliansubgroupis
containedinZ(G)|71
VI | Contents
§204 TheoremofR.vanderWaalonp-groupswithcyclicderivedsubgroup,
p>2|73
§205 MaximalsubgroupsofA2-groups|75
§206 p-groupsallofwhoseminimalnonabeliansubgroupsarepairwise
nonisomorphic|90
§207 Metacyclicgroupsofexponentpewithanormalcyclicsubgroupof
orderpe|94
§208 Non-Dedekindianp-groupsallofwhosenonnormalmaximalcyclic
subgroupsaremaximalabelian|99
§209 p-groupswithmanyminimalnonabeliansubgroups,3|101
§210 AgeneralizationofDedekindiangroups|103
§211 Nonabelianp-groupsgeneratedbythecentersoftheirmaximal
subgroups|108
§212 Nonabelianp-groupsgeneratedbyanytwononconjugatemaximalabelian
subgroups|110
§213 p-groupswithA∩BbeingmaximalinAorBforanytwononincident
subgroupsAandB|112
§214 Nonabelianp-groupswithasmallnumberofnormalsubgroups|117
§215 Everyp-groupofmaximalclassandorder≥pp,p>3,hasexactlyp
two-generatornonabeliansubgroupsofindexp|120
§216 OnthetheoremofMannaboutp-groupsallofwhosenonnormalsubgroups
areelementaryabelian|122
§217 Nonabelianp-groupsallofwhoseelementscontainedinanyminimal
nonabeliansubgroupareofbreadth<2|129
§218 Anonabeliantwo-generatorp-groupinwhichanynonabelianepimorphic
imagehasthecycliccenter|130
Contents | VII
§219 On“large”elementaryabeliansubgroupsinp-groupsofmaximal
class|132
§220 Onmetacyclicp-groupsandclosetothem|136
§221 Non-Dedekindianp-groupsinwhichnormalclosuresofnonnormalabelian
subgroupshavecycliccenters|141
§222 CharacterizationofDedekindianp-groups,2|143
§223 Non-Dedekindianp-groupsinwhichthenormalclosureofanynonnormal
cyclicsubgroupisnonabelian|147
§224 p-groupsinwhichthenormalclosureofanycyclicsubgroupis
abelian|154
§225 Nonabelianp-groupsinwhichanys(afixeds∈{3,...,p+1})pairwise
noncommutingelementsgenerateagroupofmaximalclass|156
§226 Noncyclicp-groupscontainingonlyonepropernormalsubgroupofagiven
order|158
§227 p-groupsallofwhoseminimalnonabeliansubgroupshavecyclic
centralizers|161
§228 Propertiesofmetahamiltonianp-groups|163
§229 p-groupsallofwhosecyclicsubgroupsoforder≥p3arenormal|170
§230 Nonabelianp-groupsofexponentpeallofwhosecyclicsubgroupsoforder
pearenormal|179
§231 p-groupswhicharenotgeneratedbytheirnonnormalsubgroups|185
§232 Nonabelianp-groupsinwhichanynonabeliansubgroupcontainsits
centralizer|191
§233 Onmonotonep-groups|194
§234 p-groupsallofwhosemaximalnonnormalabeliansubgroupsare
conjugate|196
VIII | Contents
§235 Onnormalsubgroupsofcapable2-groups|197
§236 Non-Dedekindianp-groupsinwhichthenormalclosureofanycyclic
subgrouphasacycliccenter|198
§237 Noncyclicp-groupsallofwhosenonnormalmaximalcyclicsubgroupsare
self-centralizing|199
§238 Nonabelianp-groupsallofwhosenonabeliansubgroupshaveacyclic
center|200
§239 p-groupsGallofwhosecyclicsubgroupsareeithercontainedinZ(G)or
avoidZ(G)|202
§240 p-groupsGallofwhosenonnormalmaximalcyclicsubgroupsare
conjugate|203
§241 Non-Dedekindianp-groupswithanormalintersectionofanytwo
nonincidentsubgroups|205
§242 Non-Dedekindianp-groupsinwhichthenormalclosuresofallnonnormal
subgroupscoincide|207
§243 Nonabelianp-groupsGwithΦ(H)=H󸀠forallnonabelianH≤G|210
§244 p-groupsinwhichanytwodistinctmaximalnonnormalsubgroupsintersect
inasubgroupoforder≤p|211
§245 On2-groupssaturatedbynonabelianDedekindiansubgroups|212
§246 Non-Dedekindianp-groupswithmanynormalsubgroups|226
§247 Nonabelianp-groupsallofwhosemetacyclicsectionsareabelian|227
§248 Non-Dedekindianp-groupsGsuchthatHG =HZ(G)forallnonnormal
H<G|228
§249 Nonabelianp-groupsGwithA∩B=Z(G)foranytwodistinctmaximal
abeliansubgroupsAandB|229
§250 Onthenumberofminimalnonabeliansubgroupsinanonabelian
p-group|230
Contents | IX
§251 p-groupsallofwhoseminimalnonabeliansubgroupsareisolated|236
§252 Nonabelianp-groupsallofwhosemaximalabeliansubgroupsare
isolated|242
§253 Maximalabeliansubgroupsofp-groups,2|244
§254 Onp-groupswithmanyisolatedmaximalabeliansubgroups|246
§255 Maximalabeliansubgroupsofp-groups,3|248
§256 AproblemofD.R.Hughesfor3-groups|249
Appendix58 AlternateproofofPassman’sTheorem1.23|251
Appendix59 Iwasawa’stheoremonmodularp-groups|253
Appendix60 Onp-groups,containingonlyonenoncyclicsubgroupoforderpe,
e≥3|255
Appendix61 Anecessaryandsufficientconditionforap-groupGtosatisfy
Φ(G)≤Z(G)|257
Appendix62 Subgroupsofsomep-groups|259
Appendix63 Intersectionsofnonnormalcyclicsubgroups|262
Appendix64 Someremarksonp-groupsallofwhosenonnormalsubgroupsare
abelian|263
Appendix65 Onp-groupsGwith|Ω1(G)|=pn|264
Appendix66 Metacyclicp-groupscontaininganabeliansubgroupof
indexp|265
Appendix67 p-groupsinwhichtheintersectionofalltheirsubgroupsoforder
p2hasorderp|267
Appendix68 The2-groupsallofwhosenonabeliantwo-generatorsubgroupsare
minimalnonabelian|270
Appendix69 SupplementtoTheorem200.1|272
X | Contents
Appendix70 Nonabelianp-groupsallofwhosemaximalcyclicsubgroups
coincidewiththeircentralizers|273
Appendix71 FinitegroupsGcontainingap-subgrouppermutablewithall
Sylowq-subgroupsofGforallq≠p|274
Appendix72 Nonabelianp-groupswithanabeliansubgroupofindexpcovered
byminimalnonabeliansubgroups|278
Appendix73 Onmetacyclicandmodularp-groups|279
Appendix74 Onp-groups,p>2,withoutsubgroupsisomorphic
toS(p3)|282
Appendix75 Irregularp-groupswith<pabsolutelyregularsubgroupsof
maximalpossibleorder|284
Appendix76 Onaclassofp-groups|286
Appendix77 Thep-groups,p>2,containingonlyonesubgroup
≅S(p3)|288
Appendix78 Furthercriterionofp-nilpotenceandπ-nilpotence|290
Appendix79 2-groupsGcontaininganonabelianmetacyclicsubgroupHof
order22eandexponent2esuchthatNG(H)ismetacyclic|292
Appendix80 Onminimalnonabeliangroupsoforder22nand
exponent2n|294
Appendix81 Onp-groupswithacyclicsubgroupofindexp3|296
Appendix82 Onnonmodularp-groupsallofwhosesubgroupsoforder>pare
quasinormal|298
Appendix83 Nonabelianregularp-groupsofexponentpearegeneratedby
minimalnonabeliansubgroupsofexponentpe|300
Appendix84 Noncyclic2-groupsinwhichallcyclicsubgroupsofanyequalorder
>2areconjugate|302
Appendix85 Onp-groupswithgivenepimorphicimages|303