Berkovich / Janko Groups of Prime Power Order

Volume 2
1. Auflage 2008
ISBN: 978-3-11-020823-8
Verlag: De Gruyter
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 47, 611 Seiten

Reihe: De Gruyter Expositions in Mathematics

This is the second of three volumes devoted to elementary finite -group theory. Similar to the first volume, hundreds of important results are analyzed and, in many cases, simplified. Important topics presented in this monograph include: (a) classification of -groups all of whose cyclic subgroups of composite orders are normal, (b) classification of 2-groups with exactly three involutions, (c) two proofs of Ward's theorem on quaternion-free groups, (d) 2-groups with small centralizers of an involution, (e) classification of 2-groups with exactly four cyclic subgroups of order 2 > 2, (f) two new proofs of Blackburn's theorem on minimal nonmetacyclic groups, (g) classification of -groups all of whose subgroups of index are abelian, (h) classification of 2-groups all of whose minimal nonabelian subgroups have order 8, (i) -groups with cyclic subgroups of index are classified.

This volume contains hundreds of original exercises (with all difficult exercises being solved) and an extended list of about 700 open problems. The book is based on Volume 1, and it is suitable for researchers and graduate students of mathematics with a modest background on algebra.

Zielgruppe

Researchers, Graduate Students of Mathematics; Academic Libraries

Autoren/Hrsg.

Berkovich, Yakov

Janko, Zvonimir

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Zahlentheorie

Inhaltsverzeichnis

1;Frontmatter;1
2;Contents;5
3;List of definitions and notations;8
4;Preface;14
5;§46. Degrees of irreducible characters of Suzuki p-groups;17
6;§47. On the number of metacyclic epimorphic images of finite p-groups;30
7;§48. On 2-groups with small centralizer of an involution, I;35
8;§49. On 2-groups with small centralizer of an involution, II;44
9;§50. Janko’s theorem on 2-groups without normal elementary abelian subgroups of order 8;59
10;§51. 2-groups with self centralizing subgroup isomorphic to E8;68
11;§52. 2-groups with 2-subgroup of small order;91
12;§53. 2-groups G with c2(G) = 4;112
13;§54. 2-groups G with cn(G) = 4, n > 2;125
14;§55. 2-groups G with small subgroup (x . G | o(x) = 2");138
15;§56. Theorem of Ward on quaternion-free 2-groups;150
16;§57. Nonabelian 2-groups all of whose minimal nonabelian subgroups are isomorphic and have exponent 4;156
17;§58. Non-Dedekindian p-groups all of whose nonnormal subgroups of the same order are conjugate;163
18;§59. p-groups with few nonnormal subgroups;166
19;§60. The structure of the Burnside group of order 212;167
20;§61. Groups of exponent 4 generated by three involutions;179
21;§62. Groups with large normal closures of nonnormal cyclic subgroups;185
22;§63. Groups all of whose cyclic subgroups of composite orders are normal;188
23;§64. p-groups generated by elements of given order;195
24;§65. A2-groups;204
25;§66. A new proof of Blackburn’s theorem on minimal nonmetacyclic 2-groups;213
26;§67. Determination of U2-groups;218
27;§68. Characterization of groups of prime exponent;222
28;§69. Elementary proofs of some Blackburn’s theorems;225
29;§70. Non-2-generator p-groups all of whose maximal subgroups are 2-generator;230
30;§71. Determination of A2-groups;249
31;§72. An-groups, n > 2;264
32;§73. Classification of modular p-groups;273
33;§74. p-groups with a cyclic subgroup of index p2;290
34;§75. Elements of order = 4 in p-groups;293
35;§76. p-groups with few A1-subgroups;298
36;§77. 2-groups with a self-centralizing abelian subgroup of type (4, 2);332
37;§78. Minimal nonmodular p-groups;339
38;§79. Nonmodular quaternion-free 2-groups;350
39;§80. Minimal non-quaternion-free 2-groups;372
40;§81. Maximal abelian subgroups in 2-groups;377
41;§82. A classification of 2-groups with exactly three involutions;384
42;§83. p-groups G with O2(G) or O2*(G) extraspecial;412
43;§84. 2-groups whose nonmetacyclic subgroups are generated by involutions;415
44;§85. 2-groups with a nonabelian Frattini subgroup of order 16;418
45;§86. p-groups G with metacyclic O2*(G);422
46;§87. 2-groups with exactly one nonmetacyclic maximal subgroup;428
47;§88. Hall chains in normal subgroups of p-groups;453
48;§89. 2-groups with exactly six cyclic subgroups of order 4;470
49;§90. Nonabelian 2-groups all of whose minimal nonabelian subgroups are of order 8;479
50;§91. Maximal abelian subgroups of p-groups;483
51;§92. On minimal nonabelian subgroups of p-groups;490
52;Appendix 16. Some central products;501
53;Appendix 17. Alternate proofs of characterization theorems of Miller and Janko on 2-groups, and some related results;508
54;Appendix 18. Replacement theorems;517
55;Appendix 19. New proof of Ward’s theorem on quaternion-free 2-groups;522
56;Appendix 20. Some remarks on automorphisms;525
57;Appendix 21. Isaacs’ examples;528
58;Appendix 22. Minimal nonnilpotent groups;532
59;Appendix 23. Groups all of whose noncentral conjugacy classes have the same size;535
60;Appendix 24. On modular 2-groups;538
61;Appendix 25. Schreier’s inequality for p-groups;542
62;Appendix 26. p-groups all of whose nonabelian maximal subgroups are either absolutely regular or of maximal class;545
63;Research problems and themes II;547
64;Backmatter;585

Über Autor(innen)

Berkovich, Yakov
Yakov Berkovich, University of Haifa, Israel

Janko, Zvonimir
Zvonimir Janko, Universität Heidelberg