De Gruyter Expositions in Mathematics Groups of Prime Power Order

410 Pages·2018·2.458 MB·English

by  Yakov G. Berkovich

YakovG.BerkovichandZvonimirJanko
GroupsofPrimePowerOrder
De Gruyter Expositions in
Mathematics
|
Editedby
LevBirbrair,Fortaleza,Brazil
VictorP.Maslov,Moscow,Russia
WalterD.Neumann,NewYorkCity,NewYork,USA
MarkusJ.Pflaum,Boulder,Colorado,USA
DierkSchleicher,Bremen,Germany
KatrinWendland,Freiburg,Germany
Volume 65
Yakov G. Berkovich and Zvonimir Janko
Groups of Prime
Power Order
|
Volume 6
MathematicsSubjectClassification2010
20-02,20D15,20E07
Authors
Prof.Dr.YakovG.Berkovich
18251Afula
Israel
[email protected]
Prof.Dr.ZvonimirJanko
Ruprecht-Karls-UniversitätHeidelberg
MathematischesInstitut
ImNeuenheimerFeld288
69120Heidelberg
Germany
[email protected]
ISBN978-3-11-053097-1
e-ISBN(PDF)978-3-11-053314-9
e-ISBN(EPUB)978-3-11-053100-8
ISSN0938-6572
LibraryofCongressControlNumber:2018941447
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2018WalterdeGruyterGmbH,Berlin/Boston
Typesetting:Dimler&Albroscheit,Müncheberg
Printingandbinding:CPIbooksGmbH,Leck
www.degruyter.com
Contents
Listofdefinitionsandnotations XIII
|
Preface XIX
|
§257 Nonabelianp-groupswithexactlyoneminimalnonabeliansubgroupof
exponent>p|1
§258 2-groupswithsomeprescribedminimalnonabeliansubgroups 6
|
§259 Nonabelianp-groups,p>2,allofwhoseminimalnonabeliansubgroups
areisomorphictoM 13
p3 |
§260 p-groupswithmanymodularsubgroupsMpn |15
§261 Nonabelianp-groupsofexponent>pwithasmallnumberofmaximal
abeliansubgroupsofexponent>p|18
§262 Nonabelianp-groupsallofwhosesubgroupsarepowerful|24
§263 Nonabelian2-groupsGwithCG(x)≤HforallH∈Γ1andx ∈H−Z(G)|25
§264 Nonabelian2-groupsofexponent≥16allofwhoseminimalnonabelian
subgroups,exceptone,haveorder8 26
|
§265 p-groupsallofwhoseregularsubgroupsareeitherabsolutelyregularorof
exponentp|29
§266 Nonabelianp-groupsinwhichanytwodistinctminimalnonabelian
subgroupswithanontrivialintersectionarenon-isomorphic 33
|
§267 Thompson’sA×Blemma|35
§268 Onautomorphismsofsomep-groups|36
§269 Oncriticalsubgroupsofp-groups|47
§270 p-groupsallofwhoseAk-subgroupsforafixedk >1aremetacyclic|50
§271 TwotheoremsofBlackburn 55
|
https://doi.org/10.1515/9783110533149-202
VI
| Contents
§272 Nonabelianp-groupsallofwhosemaximalabeliansubgroups,exceptone,
areeithercyclicorelementaryabelian 57
|
§273 Nonabelianp-groupsallofwhosenoncyclicmaximalabeliansubgroupsare
elementaryabelian 58
|
§274 Non-Dedekindianp-groupsinwhichanytwononnormalsubgroups
normalizeeachother 59
|
§275 Nonabelianp-groupswithexactlypnormalclosuresofminimalnonabelian
subgroups 61
|
§276 2-groupsallofwhosemaximalsubgroups,exceptone,are
Dedekindian 63
|
§277 p-groupswithexactlytwoconjugateclassesofnonnormalmaximalcyclic
subgroups 67
|
§278 Nonmetacyclicp-groupsallofwhosemaximalmetacyclicsubgroupshave
indexp|68
§279 Subgroupcharacterizationofsomep-groupsofmaximalclassandcloseto
them 70
|
§280 Nonabelianp-groupsallofwhosemaximalsubgroups,exceptone,are
minimalnonmetacyclic 73
|
§281 Nonabelianp-groupsinwhichanytwodistinctminimalnonabelian
subgroupshaveacyclicintersection 80
|
§282 p-groupswithlargenormalclosuresofnonnormalsubgroups|83
§283 Nonabelianp-groupswithmanycycliccentralizers|86
§284 Nonabelianp-groups,p>2,ofexponent>p2allofwhoseminimal
nonabeliansubgroupsareoforderp3|87
§285 AgeneralizationofLemma57.1 88
|
§286 Groupsofexponentpwithmanynormalsubgroups|90
VII
Contents |
§287 p-groupsinwhichtheintersectionofanytwononincidentsubgroupsis
normal 92
|
§288 Nonabelianp-groupsinwhichforeveryminimalnonabelianM<Gand
x ∈G−M,wehaveCM(x)=Z(M)|97
§289 Non-Dedekindianp-groupsallofwhosemaximalnonnormalsubgroupsare
conjugate 98
|
§290 Non-Dedekindianp-groupsGwithanoncyclicpropersubgroupHsuchthat
eachsubgroupwhichisnonincidentwithHisnormalinG|99
§291 Nonabelianp-groupswhicharegeneratedbyafixedmaximalcyclic
subgroupandanyminimalnonabeliansubgroup 100
|
§292 Nonabelianp-groupsgeneratedbyanytwonon-conjugateminimal
nonabeliansubgroups 101
|
§293 Exercises 102
|
§294 p-groups,p>2,whoseFrattinisubgroupisnonabelianmetacyclic|174
§295 Anyirregularp-groupcontainsanon-isolatedmaximalregular
subgroup 176
|
§296 Non-Dedekindianp-groupsallofwhosenonnormalmaximalcyclic
subgroupsareC-equivalent 178
|
§297 On2-groupswithoutelementaryabeliansubgroupoforder8 180
|
§298 Non-Dedekindianp-groupsallofwhosesubgroupsoforder≤ps(s≥1
fixed)arenormal 182
|
§299 Onp󸀠-automorphismsofp-groups|184
§300 Onp-groupsallofwhosemaximalsubgroupsofexponentparenormaland
haveorderpp|185
§301 p-groupsofexponent>pcontaining<pmaximalabeliansubgroupsof
exponent>p|188
§302 AlternateproofofTheorem109.1 190
|
VIII
| Contents
§303 Nonabelianp-groupsoforder>p4allofwhosesubgroupsoforderp4are
isomorphic 192
|
§304 Non-Dedekindianp-groupsinwhicheachnonnormalsubgrouphasacyclic
complementinitsnormalizer 196
|
§305 Nonabelianp-groupsGallofwhoseminimalnonabeliansubgroupsM
satisfyZ(M)≤Z(G)|198
§306 Nonabelian2-groupsallofwhosemaximalsubgroups,exceptone,are
quasi-Hamiltonianorabelian 200
|
§307 Nonabelianp-groups,p>2,allofwhosemaximalsubgroups,exceptone,
arequasi-Hamiltonianorabelian 205
|
§308 Nonabelianp-groupswithanelementaryabelianintersectionofanytwo
distinctmaximalabeliansubgroups 210
|
§309 Minimalnon-p-centralp-groups|211
§310 Nonabelianp-groupsinwhicheachelementinanyminimalnonabelian
subgroupishalf-central 213
|
§311 Nonabelianp-groupsGofexponentpinwhichC (x)=⟨x⟩Gforall
G
noncentralx ∈G|214
§312 Nonabelian2-groupsallofwhoseminimalnonabeliansubgroups,
exceptone,areisomorphicto
M2(2,2)=⟨a,b|a4 =b4 =1, ab =a−1⟩|215
§313 Non-Dedekindian2-groupsallofwhosemaximalDedekindiansubgroups
haveindex2 223
|
§314 TheoremofGlauberman–Mazzaonp-groupswithanonnormalmaximal
elementaryabeliansubgroupoforderp2|224
§315 p-groupswithsomenon-p-centralmaximalsubgroups|227
§316 Nonabelianp-groups,p>2,ofexponent>p3allofwhoseminimal
nonabeliansubgroups,exceptone,haveorderp3|228
IX
Contents |
§317 Nonabelianp-groups,p>2,allofwhoseminimalnonabeliansubgroups
areisomorphictoMp(2,2)|231
§318 Nonabelianp-groups,p>2,ofexponent>p2allofwhoseminimal
nonabeliansubgroups,exceptone,areisomorphictoMp(2,2)|233
§319 Anewcharacterizationofp-centralp-groups|236
§320 Nonabelianp-groupswithexactlyonenon-p-centralminimalnonabelian
subgroup 237
|
§321 Nonabelianp-groupsGinwhicheachelementinG−Φ(G)is
half-central 239
|
§322 Nonabelianp-groupsGsuchthatC (H)=Z(G)foranynonabelian
G
H≤G|240
§323 Nonabelianp-groupsthatarenotgeneratedbyitsnoncyclicabelian
subgroups 241
|
§324 Aseparationofmetacyclicandnonmetacyclicminimalnonabelian
subgroupsinnonabelianp-groups|242
§325 p-groupswhicharenotgeneratedbytheirnonnormalsubgroups,2|243
§326 Nonabelianp-groupsallofwhosemaximalabeliansubgroupsare
normal 244
|
A.110 Non-absolutelyregularp-groupsallofwhosemaximalabsolutelyregular
subgroupshaveindexp|245
A.111 Nonabelianp-groupsofexponent>pallofwhosemaximalabelian
subgroupsofexponent>pareisolated|247
A.112 Metacyclicp-groupswithanabelianmaximalsubgroup|249
A.113 Nonabelianp-groupswithacyclicintersectionofanytwodistinctmaximal
abeliansubgroups 251
|
A.114 AnanalogofThompson’sdihedrallemma 253
|
A.115 SomeresultsfromThompson’papersandtheOddOrderpaper 255
|
X
| Contents
A.116 Onnormalsubgroupsofap-group|257
A.117 TheoremofMann 263
|
A.118 Onp-groupswithgivenisolatedsubgroups|264
A.119 Two-generatornormalsubgroupsofap-groupGthatcontainedinΦ(G)are
metacyclic 269
|
A.120 Alternateproofsofsomecountingtheorems 270
|
A.121 Onp-groupsofmaximalclass|275
A.122 Criteriaofregularity 283
|
A.123 Nonabelianp-groupsinwhichanytwononincidentsubgroupshavean
abelianintersection 285
|
A.124 Characterizationsofthep-groupsofmaximalclassandtheprimary
ECF-groups 287
|
A.125 Nonabelianp-groupsallofwhosepropernonabeliansubgroupshave
exponentp|289
A.126 Onp-groupswithabelianautomorphismgroups|291
A.127 AlternateproofofProposition1.23 293
|
A.128 AlternateproofofthetheoremofPassmanonp-groupsallofwhose
subgroupsoforder≤ps(s≥1isfixed)arenormal|294
A.129 AlternateproofsofTheorems309.1and309.2onminimalnon-p-central
p-groups|297
A.130 Non-Dedekindianp-groupsallofwhosenonnormalmaximalcyclic
subgroupsareconjugate 299
|
A.131 Acharacterizationofsome3-groupsofmaximalclass 301
|
A.132 Alternateapproachtoclassificationofminimalnon-p-central
p-groups|303