Enumeration of finite groups:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2007
|
| Edition: | 1. publ. |
| Series: | Cambridge tracts in mathematics
173 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 281 S. |
| ISBN: | 9780521882170 |
Staff View
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Record in the Search Index
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| adam_text | SIMON R BLACKBURN ROYAL HOLLOWAY, UNIVERSITY OF LONDON PETER M NEUMANN
THE QUEEN S COLLEGE, OXFORD ST STEPHEN S COLLEGE, ,UNIVERSITY OF DELHI
, . ; ;* , * - - . . **** : : . : * : . , : I : * : * : * ; , ,
. ^ * * - , I . * * * * , ; !*,.-*..* * :* , ** . R , .**::**: F
* C . * ! ; - ! . I I I . J ENUMERATION OF FINITE. GROUPS
.H, J J - »*-****» .. .1 J I * I.T J IP TI . !-^. LI ( ! JO
.3II:. ! F C CAMBRIDGE UNIVERSITY PRESS CONTENTS * P R E F A C E
--. ** ** * X I 1 INTRODUCTION , . ! 1 1 ELEMENTARY RESULTS . , 3 2
SOME BASIC OBSERVATIONS 5 II GROUPS OF PRIME POWER ORDER L 9 3
PRELIMINARIES .... *;? T , 11 3.1 TENSOR PRODUCTS AND EXTERIOR SQUARES
OF ABELIAN G R O U P S : : , : - ..:*;- * * ; : ; - * * ; -
* *: - * * * * . . ; * * * * * * * * . , * * * * - 1 II
3.2 COMMUTATORS AND NILPOTENT GROUPS ;. ,,,- 12 3.3 THE FRATTINI
SUBGROUP ... . 17 3.4 LINEAR ALGEBRA; ; . ; : * *., ..»* , ^^ . 19 4
* E N U M E R A T I N G ^ - G R O U P S : A L O W E R B O U N D . ,
: ; ; 2 3 4.1. RELATIVELY FREE;GROUPS ? * . *.,:** .( . I.. . :
23 ** 4.2 PROOF OF THE LOWER BOUND O.- ; , ** UDS - 26 5 .
ENUMERATINGP-GROUPS: UPPER BOUNDS. ;:!,*-* Y . I 28 5.1 AN ELEMENTARY
UPPER BOUND 28 *V 5.2 AN OVERVIEW OF THE SIMS APPROACH * V * I; *
* : J; , .. - ; 30 I 5.3 LINEARISING THE PROBLEMS F.; . *.**.-.:
O:I 7 M . R », . . 31 * * 5^4 A SMALL SET OF RELATIONS ** .
, ? .^. IT^VUP * , *; . .* 35 * 5.5 PROOF OF THE UPPER BOUND
* . . I **I.UO T * *:. * .. 40 VU VIII CONTENTS III PYBER S THEOREM
45 6 SOME MORE PRELIMINARIES * ., *. 47 6.1 HALL SUBGROUPS AND-SYLOW
SYSTEMS - --- 47 6.2 THE FITTING SUBGROUP 50 6.3 PERMUTATIONS AND
PRIMITIVITY 52 7 GROUP EXTENSIONS AND COHOMOLOGY 60 7.1 GROUP EXTENSIONS
60 7.2 COHOMOLOGY 67 7.3 RESTRICTION AND TRANSFER 73 7.4 THE MCLVER AND
NEUMANN BOUND 75 8 SOME REPRESENTATION THEORY 78 8.1 SEMISIMPLE ALGEBRAS
*, - 78 8.2 CLIFFORD S THEOREM 80 8.3 THE SKOLEM-NOETHER THEOREM * * .**
V 81 8.4 EVERY FINITE SKEW FIELD IS A FIELD 85 9 PRIMITIVE SOLUBLE
LINEAR GROUPS . 88 9.1 SOME BASIC STRUCTURE THEORY 88 9.2 THE SUBGROUP
B. *, - , . . . ..-../,.. . 90 10 THE ORDERS OF GROUPS . : , 94 11
CONJUGACY CLASSES OF MAXIMAL SOLUBLE SUBGROUPS OF -* I SYMMETRIC GROUPS
* . ; 1 * , . ... 98 12 ENUMERATION OF FINITE GROUPS WITH ABELIAN
SYLOW: , ; , SUBGROUPS 102 12.1 COUNTING SOLUBLE A-GROUPS:.. AN
OVERVIEW . T T *. * : :, 103 -.. 12.2 SOLUBLE A-SUBGROUPS OF THE
GENERAL .LINEAR GROUP AND THE SYMMETRIC GROUPS {*:;:* -. **- : *
**. , : * 103 12.3 MAXIMAL SOLUBLE P -A-SUBGROUPS 108 12.4
ENUMERATION OF SOLUBLE A-GROUPS , ; .-, /^ S -;. . * 109- 13 MAXIMAL
SOLUBLE LINEAR GROUPS ; . ;- */, ** I-. .: . , 113 13.1 THE FIELD K
AND A SUBFIELD OF K , . , : F..-. * -. 113 :: 13.2 THE QUOTIENT G/C AND
THE ALGEBRA (C) * -X 114 * ; 13.3 THE QUOTIENT B/A
-*** :*-.:,%).: . **. ,.; C : 116 CONTENTS IX 13.4 THE SUBGROUP B
- .,.V .-*;.**;:.*. , - : 119 13.5 STRUCTURE OFG DETERMINED BY:B;
.. ;.(*. ., ; : . - * 1 125 14 CONJUGACY CLASSES OF MAXIMAL SOLUBLE
SUBGROUPS OF THE ; . * GENERAL LINEAR GROUPS 127 15 PYBER S THEOREM: THE
SOLUBLE CASE 132 15.1 EXTENSIONS AND SOLUBLE SUBGROUPS *-** . * 133 15.2
PYBER S THEOREM 135 16 PYBER S THEOREM: THE GENERAL CASE * * 140 16.1
THREE THEOREMS ON GROUP GENERATION 140 16.2 UNIVERSAL CENTRAL EXTENSIONS
AND COVERING GROUPS 146 16.3 THE GENERALISED FITTING SUBGROUP 150 16.4
THE GENERAL CASE OF PYBER S THEOREM 154 IV OTHER TOPICS 161 17
ENUMERATION WITHIN VARIETIES OF ABELIAN GROUPS , 163 17.1 VARIETIES OF
ABELIAN GROUPS 164 17.2 ENUMERATING PARTITIONS 167 17.3 FURTHER RESULTS
ON ABELIAN GROUPS 173 18 ENUMERATION WITHIN SMALL VARIETIES OF A-GROUPS
174 18.1 A MINIMAL VARIETY OF A-GROUPS 175 18.2 THE JOIN OF MINIMAL
VARIETIES * 184 19 ENUMERATION WITHIN SMALL VARIETIES OF P-GROUPS 187
19.1 ENUMERATING TWO SMALL VARIETIES 189 19.2 THE RATIO OF TWO
ENUMERATION FUNCTIONS 191 20 MISCELLANEA 195 20.1 ENUMERATING
D-GENERATOR GROUPS * . : 195 20.2 GROUPS WITH FEW RION-ABELIAH
COMPOSITION FACTORS * 206 20.3 ENUMERATING GRADED LIE RINGS 211 20.4
GROUPS OF NILPOTENCY CLASS 3 * *.:***. * 216 21 SURVEY OF OTHER RESULTS
222 21.1 GRAHAM HIGRRIAN S PORC CONJECTURE ** -- -222 21.2 ISOCLINISM
CLASSES OF P : GROUPS * * . 224 X * CONTENTS 21.3 GROUPS OF SQUARE-FREE
ORDER * V .V. .- *. 227 * 21.4 GROUPS OF CUBE-FREE ORDER -. *
*; / .:.-.. -.: 233 21.5 GROUPS OF ARITHMETICALLY SMALL ORDERS 236 21.6
SURJECTIVITY OF,THE ENUMERATION FUNCTION * , * ,; 238 V * 21.7
DENSITIES OF CERTAIN SETS OF GROUP ORDERS ...- * - 246 21.8 ENUMERATING
PERFECT GROUPS 256 22 S O M E OPEN PROBLEMS ... . : *:* -R * : -. .:..
259 APPENDIX A: MAXIMISING TWO FUNCTIONS 269 ; I:. R E F E R E N C E S
.- *** %I .:. -^ R- : *,, .. ,:**. .: *.-_ ** ^ 2 7 5 .-. INDEX
; I : *.**;* *(*:,.., 2 8 0 1 . - *;** : J .**.:- F
|
| adam_txt |
SIMON R BLACKBURN ROYAL HOLLOWAY, UNIVERSITY OF LONDON PETER M NEUMANN
THE QUEEN'S COLLEGE, OXFORD ST STEPHEN'S COLLEGE, ,UNIVERSITY OF DELHI "
, . ; ;* ' , * - - . . **** : : . : * : . , : I : * : * : * ; , ' , '
. ^ * * - , I . * * * * , ; ' !*,.-*.*"* ':*', **'. R , .**::**: F
* ' C . * ! ; - ! ' . I I ' I . J ENUMERATION OF FINITE. GROUPS
'"'.H,' J J - »*-****» "'. .1 J ' I \ * I.T'J IP"TI'.'!-^.'LI'"('! JO
.3II:. ! F C CAMBRIDGE ' UNIVERSITY PRESS CONTENTS * " P R E F A C E
--. ** '** '* X I 1 INTRODUCTION , . ! 1 1 ELEMENTARY RESULTS . , 3 2
SOME BASIC OBSERVATIONS 5 II GROUPS OF PRIME POWER ORDER L ' 9 3
PRELIMINARIES . *;? T , 11 3.1 TENSOR PRODUCTS AND EXTERIOR SQUARES
OF ABELIAN G R O U P S ' : : , : - .:*;- * * ' ; : ; - * * ; ' '" -
*'*: - ' " * * * * . . ; * * ' ' * * * * * * . ' , * * * * - 1 II
3.2 COMMUTATORS AND NILPOTENT GROUPS ;. ,,,- 12 3.3 THE FRATTINI
SUBGROUP . . 17 3.4 LINEAR ALGEBRA; ; '. ; : * *., .»* , ^^ . 19 4
* E N U M E R A T I N G ^ - G R O U P S : A L O W E R B O U N D . , '
: ; ; 2 3 4.1. RELATIVELY FREE;GROUPS ? * . *.,:** "\.( . I. . : '
23 '** 4.2 PROOF OF THE LOWER BOUND O.-' ; , ** UDS - 26 5 . '
ENUMERATINGP-GROUPS: UPPER BOUNDS. ;:!,*-* 'Y'.'I 28 5.1 AN ELEMENTARY
UPPER BOUND 28 *V ' 5.2 AN OVERVIEW OF THE SIMS APPROACH *" V '* I; *
*': J;"",' '. - ; 30 I 5.3 'LINEARISING' THE PROBLEMS F.; .'*.**.-.:
O:I 7 M . R »,' . . 31 '*'* 5^4 A SMALL SET OF RELATIONS **' '.
\,"?\'.^. IT^VUP *', *;"' '".'.* 35 '*' 5.5 PROOF OF THE UPPER BOUND
'*'.'.' I **I.UO T * *:." * . 40 VU VIII CONTENTS III PYBER'S THEOREM
45 6 SOME MORE PRELIMINARIES * ., *. 47 6.1 HALL SUBGROUPS AND-SYLOW
SYSTEMS - --- 47 6.2 THE FITTING SUBGROUP 50 6.3 PERMUTATIONS AND
PRIMITIVITY 52 7 GROUP EXTENSIONS AND COHOMOLOGY 60 7.1 GROUP EXTENSIONS
60 7.2 COHOMOLOGY 67 7.3 RESTRICTION AND TRANSFER 73 7.4 THE MCLVER AND
NEUMANN BOUND 75 8 SOME REPRESENTATION THEORY 78 8.1 SEMISIMPLE ALGEBRAS
*, - 78 8.2 CLIFFORD'S THEOREM 80 8.3 THE SKOLEM-NOETHER THEOREM * * .**
V 81 8.4 EVERY FINITE SKEW FIELD IS A FIELD 85 9 PRIMITIVE SOLUBLE
LINEAR GROUPS . 88 9.1 SOME BASIC STRUCTURE THEORY 88 9.2 THE SUBGROUP
B. *, - , . . . .-./,. . 90 10 THE ORDERS OF GROUPS . : , 94 11
CONJUGACY CLASSES OF MAXIMAL SOLUBLE SUBGROUPS OF -* I SYMMETRIC GROUPS
* . ; 1 * , . . 98 12 ENUMERATION OF FINITE GROUPS WITH ABELIAN
SYLOW: , ; ', SUBGROUPS 102 12.1 COUNTING SOLUBLE A-GROUPS:. AN
OVERVIEW . T T *. * : :, 103' '-. 12.2 SOLUBLE A-SUBGROUPS OF THE
GENERAL .LINEAR GROUP AND THE SYMMETRIC GROUPS {*:;:*'-. **- ' : '*
"**. ,'\ : * 103 12.3 MAXIMAL SOLUBLE P'-A-SUBGROUPS 108 12.4
ENUMERATION OF SOLUBLE A-GROUPS ,'; .-, /^ S -;. . * 109- 13 MAXIMAL
SOLUBLE LINEAR GROUPS ; . ;- */, '** I-. .: . , 113 ' 13.1 THE FIELD K
AND A SUBFIELD OF K , .', : F.-. * '-. 113 :: 13.2 THE QUOTIENT G/C AND
THE ALGEBRA '(C) ' ' * -X 114 *'; 13.3 THE QUOTIENT B/A
-*** :*-.:,%).: . **. ',.; " C : 116 CONTENTS IX 13.4 THE SUBGROUP B
- .,.V \ .-*;.**;:.*. , - : 119 13.5 STRUCTURE OFG DETERMINED BY:B;
. ;.(*. ., ; : . - * 1 125 14 CONJUGACY CLASSES OF MAXIMAL SOLUBLE
SUBGROUPS OF THE ;".'* GENERAL LINEAR GROUPS 127 15 PYBER'S THEOREM: THE
SOLUBLE CASE 132 15.1 EXTENSIONS AND SOLUBLE SUBGROUPS *-**'. * 133 15.2
PYBER'S THEOREM 135 16 PYBER'S THEOREM: THE GENERAL CASE *"* 140 16.1
THREE THEOREMS ON GROUP GENERATION 140 16.2 UNIVERSAL CENTRAL EXTENSIONS
AND COVERING GROUPS 146 16.3 THE GENERALISED FITTING SUBGROUP 150 16.4
THE GENERAL CASE OF PYBER'S THEOREM 154 IV OTHER TOPICS 161 17
ENUMERATION WITHIN VARIETIES OF ABELIAN GROUPS , 163 17.1 VARIETIES OF
ABELIAN GROUPS 164 17.2 ENUMERATING PARTITIONS 167 17.3 FURTHER RESULTS
ON ABELIAN GROUPS 173 18 ENUMERATION WITHIN SMALL VARIETIES OF A-GROUPS
174 18.1 A MINIMAL VARIETY OF A-GROUPS 175 18.2 THE JOIN OF MINIMAL
VARIETIES * 184 19 ENUMERATION WITHIN SMALL VARIETIES OF P-GROUPS 187
19.1 ENUMERATING TWO SMALL VARIETIES 189' 19.2 THE RATIO OF TWO
ENUMERATION FUNCTIONS 191 20 MISCELLANEA 195 20.1 ENUMERATING
D-GENERATOR'GROUPS * . : 195 20.2 GROUPS WITH FEW RION-ABELIAH
COMPOSITION FACTORS' * 206 20.3 ENUMERATING GRADED LIE RINGS 211 20.4
GROUPS OF NILPOTENCY CLASS 3 * *.:***. * 216 21 SURVEY OF OTHER RESULTS
222 21.1 GRAHAM HIGRRIAN'S PORC CONJECTURE ** --' -222 21.2 ISOCLINISM
CLASSES OF P : GROUPS '* * . 224 X * CONTENTS 21.3 GROUPS OF SQUARE-FREE
ORDER ' * V .V.'.- " *. 227 "* ' 21.4 GROUPS OF CUBE-FREE ORDER' -. *
*; / .:.-. -.: 233 21.5 GROUPS OF ARITHMETICALLY SMALL ORDERS 236 21.6
SURJECTIVITY OF,THE ENUMERATION FUNCTION * , * ,; 238 "V * 21.7
DENSITIES OF CERTAIN SETS OF GROUP ORDERS .-'* - 246 21.8 ENUMERATING
PERFECT GROUPS 256 22 S O M E OPEN PROBLEMS . . : *:* -R * : -.'.:.
259 APPENDIX A: MAXIMISING TWO FUNCTIONS 269 ; I:. R E F E R E N C E S
' .- *** %I .:.\ -^ R- : *,, . ,:**. .: *.-_"** ^ 2 7 5 .-. INDEX
; I : *.**;* *(*:,., 2 8 0 " 1 . -'*;**': J .**.:-' F |
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| author | Blackburn, Simon R. Neumann, Peter M. Venkataraman, Geetha |
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| dewey-ones | 512 - Algebra |
| dewey-raw | 512.23 |
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| dewey-sort | 3512.23 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
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| id | DE-604.BV022969719 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T19:08:03Z |
| indexdate | 2024-07-09T21:08:50Z |
| institution | BVB |
| isbn | 9780521882170 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016173988 |
| oclc_num | 254263247 |
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| owner_facet | DE-703 DE-11 |
| physical | XII, 281 S. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Cambridge tracts in mathematics |
| series2 | Cambridge tracts in mathematics |
| spelling | Blackburn, Simon R. Verfasser (DE-588)113768931 aut Enumeration of finite groups Simon R. Blackburn ; Peter M. Neumann ; Geetha Venkataraman 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2007 XII, 281 S. txt rdacontent n rdamedia nc rdacarrier Cambridge tracts in mathematics 173 Finite groups Abzählen (DE-588)4508960-7 gnd rswk-swf Endliche Gruppe (DE-588)4014651-0 gnd rswk-swf Endliche Gruppe (DE-588)4014651-0 s Abzählen (DE-588)4508960-7 s DE-604 Neumann, Peter M. Verfasser aut Venkataraman, Geetha Verfasser aut Cambridge tracts in mathematics 173 (DE-604)BV000000001 173 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016173988&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Blackburn, Simon R. Neumann, Peter M. Venkataraman, Geetha Enumeration of finite groups Cambridge tracts in mathematics Finite groups Abzählen (DE-588)4508960-7 gnd Endliche Gruppe (DE-588)4014651-0 gnd |
| subject_GND | (DE-588)4508960-7 (DE-588)4014651-0 |
| title | Enumeration of finite groups |
| title_auth | Enumeration of finite groups |
| title_exact_search | Enumeration of finite groups |
| title_exact_search_txtP | Enumeration of finite groups |
| title_full | Enumeration of finite groups Simon R. Blackburn ; Peter M. Neumann ; Geetha Venkataraman |
| title_fullStr | Enumeration of finite groups Simon R. Blackburn ; Peter M. Neumann ; Geetha Venkataraman |
| title_full_unstemmed | Enumeration of finite groups Simon R. Blackburn ; Peter M. Neumann ; Geetha Venkataraman |
| title_short | Enumeration of finite groups |
| title_sort | enumeration of finite groups |
| topic | Finite groups Abzählen (DE-588)4508960-7 gnd Endliche Gruppe (DE-588)4014651-0 gnd |
| topic_facet | Finite groups Abzählen Endliche Gruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016173988&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000001 |
| work_keys_str_mv | AT blackburnsimonr enumerationoffinitegroups AT neumannpeterm enumerationoffinitegroups AT venkataramangeetha enumerationoffinitegroups |