Description
This is the first of three volumes of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this monograph include: (a) counting of subgroups, with almost all main counting theorems being proved, (b) regular p-groups and regularity criteria, (c) p-groups of maximal class and their numerous characterizations, (d) characters of p-groups, (e) p-groups with large Schur multiplier and commutator subgroups, (f) (p‒1)-admissible Hall chains in normal subgroups, (g) powerful p-groups, (h) automorphisms of p-groups, (i) p-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.
Chapter
§3. Minimal classes
pp.：
78 – 89
§4. p-groups with cyclic Frattini subgroup
pp.：
89 – 93
§5. Hall’s enumeration principle
pp.：
93 – 101
§6. q'-automorphisms of q-groups
pp.：
101 – 111
§7. Regular p-groups
pp.：
111 – 118
§8. Pyramidal p-groups
pp.：
118 – 129
§9. On p-groups of maximal class
pp.：
129 – 134
§10. On abelian subgroups of p-groups
pp.：
134 – 148
§11. On the power structure of a p-group
pp.：
148 – 166
§12. Counting theorems for p-groups of maximal class
pp.：
166 – 171
§13. Further counting theorems
pp.：
171 – 181
§14. Thompson’s critical subgroup
pp.：
181 – 205
§15. Generators of p-groups
pp.：
205 – 209
§16. Classification of finite p-groups all of whose noncyclic subgroups are normal
pp.：
209 – 212
§17. Counting theorems for regular p-groups
pp.：
212 – 218
§18. Counting theorems for irregular p-groups
pp.：
218 – 222
§19. Some additional counting theorems
pp.：
222 – 235
§20. Groups with small abelian subgroups and partitions
pp.：
235 – 239
§21. On the Schur multiplier and the commutator subgroup
pp.：
239 – 242
§22. On characters of p-groups
pp.：
242 – 249
§23. On subgroups of given exponent
pp.：
249 – 262
§24. Hall’s theorem on normal subgroups of given exponent
pp.：
262 – 266
§25. On the lattice of subgroups of a group
pp.：
266 – 276
§26. Powerful p-groups
pp.：
276 – 282
§27. p-groups with normal centralizers of all elements
pp.：
282 – 295
§28. p-groups with a uniqueness condition for nonnormal subgroups
pp.：
295 – 299
§29. On isoclinism
pp.：
299 – 305
§30. On p-groups with few nonabelian subgroups of order pp and exponent p
pp.：
305 – 309
§31. On p-groups with small p0-groups of operators
pp.：
309 – 321
§32. W. Gaschütz’s and P. Schmid’s theorems on p-automorphisms of p-groups
pp.：
321 – 329
§33. Groups of order pm with automorphisms of order pm-1, pm-2 or pm-3
pp.：
329 – 334
§34. Nilpotent groups of automorphisms
pp.：
334 – 338
§35. Maximal abelian subgroups of p-groups
pp.：
338 – 346
§36. Short proofs of some basic characterization theorems of finite p-group theory
pp.：
346 – 353
§37. MacWilliams’ theorem
pp.：
353 – 365
§38. p-groups with exactly two conjugate classes of subgroups of small orders and exponentp > 2
pp.：
365 – 368
§39. Alperin’s problem on abelian subgroups of small index
pp.：
368 – 371
§40. On breadth and class number of p-groups
pp.：
371 – 375
§41. Groups in which every two noncyclic subgroups of the same order have the same rank
pp.：
375 – 378
§42. On intersections of some subgroups
pp.：
378 – 382
§43. On 2-groups with few cyclic subgroups of given order
pp.：
382 – 385
§44. Some characterizations of metacyclic p-groups
pp.：
385 – 392
§45. A counting theorem for p-groups of odd order
pp.：
392 – 397
Appendix 1. The Hall–Petrescu formula
pp.：
397 – 399
Appendix 2. Mann’s proof of monomiality of p-groups
pp.：
399 – 403
Appendix 3. Theorems of Isaacs on actions of groups
pp.：
403 – 405
Appendix 4. Freiman’s number-theoretical theorems
pp.：
405 – 413
Appendix 5. Another proof of Theorem 5.4
pp.：
413 – 419
Appendix 6. On the order of p-groups of given derived length
pp.：
419 – 421
Appendix 7. Relative indices of elements of p-groups
pp.：
421 – 425
Appendix 8. p-groups withabsolutely regular Frattini subgroup
pp.：
425 – 429
Appendix 9. On characteristic subgroups of metacyclic groups
pp.：
429 – 432
Appendix 10. On minimal characters of p-groups
pp.：
432 – 437
Appendix 11. On sums of degrees of irreducible characters
pp.：
437 – 439
Appendix 12. 2-groups whose maximal cyclic subgroups of order > 2 are self-centralizing
pp.：
439 – 442
Appendix 13. Normalizers of Sylow p-subgroups of symmetric groups
pp.：
442 – 445
Appendix 14. 2-groups with an involution contained in only one subgroup of order 4
pp.：
445 – 451
Appendix 15. A criterion for a group to be nilpotent
pp.：
451 – 453
Research problems and themes I
pp.：
453 – 457
Backmatter
pp.：
457 – 500
Yakov Berkovich; Zvonimir Janko: Groups of Prime Power Order. Volume 1
( De Gruyter Expositions in Mathematics )
Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.
