Groups of Prime Power Order Volume 1

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