Volume 1

Yakov Berkovich

Published by De Gruyter 2008

Volume 1

Yakov Berkovich

Part of the multi-volume work Groups of Prime Power Order

Volume 46 in the series De Gruyter Expositions in Mathematics

https://doi.org/10.1515/9783110208221

Overview
Contents

About this book

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.

Author / Editor information

Yakov Berkovich , University of Haifa, Israel.

Topics

Frontmatter Publicly Available Download PDF	i
Contents Publicly Available Download PDF	v
List of definitions and notations Publicly Available Download PDF	ix
Foreword Publicly Available Download PDF	xv
Preface Publicly Available Download PDF	xvii
Introduction Requires Authentication Unlicensed Licensed Download PDF	1
§1. Groups with a cyclic subgroup of index p. Frattini subgroup. Varia Requires Authentication Unlicensed Licensed Download PDF	22
§2. The class number, character degrees Requires Authentication Unlicensed Licensed Download PDF	58
§3. Minimal classes Requires Authentication Unlicensed Licensed Download PDF	69
§4. p-groups with cyclic Frattini subgroup Requires Authentication Unlicensed Licensed Download PDF	73
§5. Hall’s enumeration principle Requires Authentication Unlicensed Licensed Download PDF	81
§6. q'-automorphisms of q-groups Requires Authentication Unlicensed Licensed Download PDF	91
§7. Regular p-groups Requires Authentication Unlicensed Licensed Download PDF	98
§8. Pyramidal p-groups Requires Authentication Unlicensed Licensed Download PDF	109
§9. On p-groups of maximal class Requires Authentication Unlicensed Licensed Download PDF	114
§10. On abelian subgroups of p-groups Requires Authentication Unlicensed Licensed Download PDF	128
§11. On the power structure of a p-group Requires Authentication Unlicensed Licensed Download PDF	146
§12. Counting theorems for p-groups of maximal class Requires Authentication Unlicensed Licensed Download PDF	151
§13. Further counting theorems Requires Authentication Unlicensed Licensed Download PDF	161
§14. Thompson’s critical subgroup Requires Authentication Unlicensed Licensed Download PDF	185
§15. Generators of p-groups Requires Authentication Unlicensed Licensed Download PDF	189
§16. Classification of finite p-groups all of whose noncyclic subgroups are normal Requires Authentication Unlicensed Licensed Download PDF	192
§17. Counting theorems for regular p-groups Requires Authentication Unlicensed Licensed Download PDF	198
§18. Counting theorems for irregular p-groups Requires Authentication Unlicensed Licensed Download PDF	202
§19. Some additional counting theorems Requires Authentication Unlicensed Licensed Download PDF	215
§20. Groups with small abelian subgroups and partitions Requires Authentication Unlicensed Licensed Download PDF	219
§21. On the Schur multiplier and the commutator subgroup Requires Authentication Unlicensed Licensed Download PDF	222
§22. On characters of p-groups Requires Authentication Unlicensed Licensed Download PDF	229
§23. On subgroups of given exponent Requires Authentication Unlicensed Licensed Download PDF	242
§24. Hall’s theorem on normal subgroups of given exponent Requires Authentication Unlicensed Licensed Download PDF	246
§25. On the lattice of subgroups of a group Requires Authentication Unlicensed Licensed Download PDF	256
§26. Powerful p-groups Requires Authentication Unlicensed Licensed Download PDF	262
§27. p-groups with normal centralizers of all elements Requires Authentication Unlicensed Licensed Download PDF	275
§28. p-groups with a uniqueness condition for nonnormal subgroups Requires Authentication Unlicensed Licensed Download PDF	279
§29. On isoclinism Requires Authentication Unlicensed Licensed Download PDF	285
§30. On p-groups with few nonabelian subgroups of order pp and exponent p Requires Authentication Unlicensed Licensed Download PDF	289
§31. On p-groups with small p0-groups of operators Requires Authentication Unlicensed Licensed Download PDF	301
§32. W. Gaschütz’s and P. Schmid’s theorems on p-automorphisms of p-groups Requires Authentication Unlicensed Licensed Download PDF	309
§33. Groups of order pm with automorphisms of order pm-1, pm-2 or pm-3 Requires Authentication Unlicensed Licensed Download PDF	314
§34. Nilpotent groups of automorphisms Requires Authentication Unlicensed Licensed Download PDF	318
§35. Maximal abelian subgroups of p-groups Requires Authentication Unlicensed Licensed Download PDF	326
§36. Short proofs of some basic characterization theorems of finite p-group theory Requires Authentication Unlicensed Licensed Download PDF	333
§37. MacWilliams’ theorem Requires Authentication Unlicensed Licensed Download PDF	345
§38. p-groups with exactly two conjugate classes of subgroups of small orders and exponentp > 2 Requires Authentication Unlicensed Licensed Download PDF	348
§39. Alperin’s problem on abelian subgroups of small index Requires Authentication Unlicensed Licensed Download PDF	351
§40. On breadth and class number of p-groups Requires Authentication Unlicensed Licensed Download PDF	355
§41. Groups in which every two noncyclic subgroups of the same order have the same rank Requires Authentication Unlicensed Licensed Download PDF	358
§42. On intersections of some subgroups Requires Authentication Unlicensed Licensed Download PDF	362
§43. On 2-groups with few cyclic subgroups of given order Requires Authentication Unlicensed Licensed Download PDF	365
§44. Some characterizations of metacyclic p-groups Requires Authentication Unlicensed Licensed Download PDF	372
§45. A counting theorem for p-groups of odd order Requires Authentication Unlicensed Licensed Download PDF	377
Appendix 1. The Hall–Petrescu formula Requires Authentication Unlicensed Licensed Download PDF	379
Appendix 2. Mann’s proof of monomiality of p-groups Requires Authentication Unlicensed Licensed Download PDF	383
Appendix 3. Theorems of Isaacs on actions of groups Requires Authentication Unlicensed Licensed Download PDF	385
Appendix 4. Freiman’s number-theoretical theorems Requires Authentication Unlicensed Licensed Download PDF	393
Appendix 5. Another proof of Theorem 5.4 Requires Authentication Unlicensed Licensed Download PDF	399
Appendix 6. On the order of p-groups of given derived length Requires Authentication Unlicensed Licensed Download PDF	401
Appendix 7. Relative indices of elements of p-groups Requires Authentication Unlicensed Licensed Download PDF	405
Appendix 8. p-groups withabsolutely regular Frattini subgroup Requires Authentication Unlicensed Licensed Download PDF	409
Appendix 9. On characteristic subgroups of metacyclic groups Requires Authentication Unlicensed Licensed Download PDF	412
Appendix 10. On minimal characters of p-groups Requires Authentication Unlicensed Licensed Download PDF	417
Appendix 11. On sums of degrees of irreducible characters Requires Authentication Unlicensed Licensed Download PDF	419
Appendix 12. 2-groups whose maximal cyclic subgroups of order > 2 are self-centralizing Requires Authentication Unlicensed Licensed Download PDF	422
Appendix 13. Normalizers of Sylow p-subgroups of symmetric groups Requires Authentication Unlicensed Licensed Download PDF	425
Appendix 14. 2-groups with an involution contained in only one subgroup of order 4 Requires Authentication Unlicensed Licensed Download PDF	431
Appendix 15. A criterion for a group to be nilpotent Requires Authentication Unlicensed Licensed Download PDF	433
Research problems and themes I Requires Authentication Unlicensed Licensed Download PDF	437
Backmatter Requires Authentication Unlicensed Licensed Download PDF	480

(Deutsch)

Language: English
Publisher: De Gruyter
Audience: Researchers, Graduate Students of Mathematics; Academic Libraries
Pages
- Front matter: 20
- Main content: 512
Keywords: Group Theory; Order; Primes

eBook

Published: December 10, 2008
ISBN: 9783110208221

Hardcover

Published: November 17, 2008
ISBN: 9783110204186