Berkovich Groups of Prime Power Order

Volume 1
1. Auflage 2008
ISBN: 978-3-11-020822-1
Verlag: De Gruyter
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 46, 532 Seiten

Reihe: De Gruyter Expositions in MathematicsISSN

This is the first of three volumes of a comprehensive and elementary treatment of finite -group theory. Topics covered in this monograph include: (a) counting of subgroups, with almost all main counting theorems being proved, (b) regular -groups and regularity criteria, (c)-groups of maximal class and their numerous characterizations, (d) characters of -groups, (e) -groups with large Schur multiplier and commutator subgroups, (f) (?1)-admissible Hall chains in normal subgroups, (g) powerful -groups, (h) automorphisms of -groups, (i) -groups all of whose nonnormal subgroups are cyclic, (j) Alperin's problem on abelian subgroups of small index.

The book is suitable for researchers and graduate students of mathematics with a modest background on algebra. It also contains hundreds of original exercises (with difficult exercises being solved) and a comprehensive list of about 700 open problems.

Zielgruppe

Researchers, Graduate Students of Mathematics; Academic Libraries

Autoren/Hrsg.

Berkovich, Yakov

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Zahlentheorie

Inhaltsverzeichnis

1;Frontmatter;1
2;Contents;5
3;List of definitions and notations;9
4;Foreword;15
5;Preface;17
6;Introduction;21
7;§1. Groups with a cyclic subgroup of index p. Frattini subgroup. Varia;42
8;§2. The class number, character degrees;78
9;§3. Minimal classes;89
10;§4. p-groups with cyclic Frattini subgroup;93
11;§5. Hall’s enumeration principle;101
12;§6. q'-automorphisms of q-groups;111
13;§7. Regular p-groups;118
14;§8. Pyramidal p-groups;129
15;§9. On p-groups of maximal class;134
16;§10. On abelian subgroups of p-groups;148
17;§11. On the power structure of a p-group;166
18;§12. Counting theorems for p-groups of maximal class;171
19;§13. Further counting theorems;181
20;§14. Thompson’s critical subgroup;205
21;§15. Generators of p-groups;209
22;§16. Classification of finite p-groups all of whose noncyclic subgroups are normal;212
23;§17. Counting theorems for regular p-groups;218
24;§18. Counting theorems for irregular p-groups;222
25;§19. Some additional counting theorems;235
26;§20. Groups with small abelian subgroups and partitions;239
27;§21. On the Schur multiplier and the commutator subgroup;242
28;§22. On characters of p-groups;249
29;§23. On subgroups of given exponent;262
30;§24. Hall’s theorem on normal subgroups of given exponent;266
31;§25. On the lattice of subgroups of a group;276
32;§26. Powerful p-groups;282
33;§27. p-groups with normal centralizers of all elements;295
34;§28. p-groups with a uniqueness condition for nonnormal subgroups;299
35;§29. On isoclinism;305
36;§30. On p-groups with few nonabelian subgroups of order pp and exponent p;309
37;§31. On p-groups with small p0-groups of operators;321
38;§32. W. Gaschütz’s and P. Schmid’s theorems on p-automorphisms of p-groups;329
39;§33. Groups of order pm with automorphisms of order pm-1, pm-2 or pm-3;334
40;§34. Nilpotent groups of automorphisms;338
41;§35. Maximal abelian subgroups of p-groups;346
42;§36. Short proofs of some basic characterization theorems of finite p-group theory;353
43;§37. MacWilliams’ theorem;365
44;§38. p-groups with exactly two conjugate classes of subgroups of small orders and exponentp > 2;368
45;§39. Alperin’s problem on abelian subgroups of small index;371
46;§40. On breadth and class number of p-groups;375
47;§41. Groups in which every two noncyclic subgroups of the same order have the same rank;378
48;§42. On intersections of some subgroups;382
49;§43. On 2-groups with few cyclic subgroups of given order;385
50;§44. Some characterizations of metacyclic p-groups;392
51;§45. A counting theorem for p-groups of odd order;397
52;Appendix 1. The Hall–Petrescu formula;399
53;Appendix 2. Mann’s proof of monomiality of p-groups;403
54;Appendix 3. Theorems of Isaacs on actions of groups;405
55;Appendix 4. Freiman’s number-theoretical theorems;413
56;Appendix 5. Another proof of Theorem 5.4;419
57;Appendix 6. On the order of p-groups of given derived length;421
58;Appendix 7. Relative indices of elements of p-groups;425
59;Appendix 8. p-groups withabsolutely regular Frattini subgroup;429
60;Appendix 9. On characteristic subgroups of metacyclic groups;432
61;Appendix 10. On minimal characters of p-groups;437
62;Appendix 11. On sums of degrees of irreducible characters;439
63;Appendix 12. 2-groups whose maximal cyclic subgroups of order > 2 are self-centralizing;442
64;Appendix 13. Normalizers of Sylow p-subgroups of symmetric groups;445
65;Appendix 14. 2-groups with an involution contained in only one subgroup of order 4;451
66;Appendix 15. A criterion for a group to be nilpotent;453
67;Research problems and themes I;457
68;Backmatter;500

Über Autor(innen)

Berkovich, Yakov
Yakov Berkovich, University of Haifa, Israel