Groups of Prime Power Order. Volume 1 (eBook)

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Yakov Berkovich , University of Haifa, Israel.

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