Groups of Prime Power Order. Volume 1

Yakov Berkovich

• Publisher: De Gruyter
• Year: 2008
• Print ISBN: 9783110204186
• eBook ISBN: 9783110208221
• Edition: 1
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

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.

Table of contents

• Frontmatter
• Contents
• List of definitions and notations
• Foreword
• Preface
• Introduction
• §1. Groups with a cyclic subgroup of index p. Frattini subgroup. Varia
• §2. The class number, character degrees
• §3. Minimal classes
• §4. p-groups with cyclic Frattini subgroup
• §5. Hall’s enumeration principle
• §6. q'-automorphisms of q-groups
• §7. Regular p-groups
• §8. Pyramidal p-groups
• §9. On p-groups of maximal class
• §10. On abelian subgroups of p-groups
• §11. On the power structure of a p-group
• §12. Counting theorems for p-groups of maximal class
• §13. Further counting theorems
• §14. Thompson’s critical subgroup
• §15. Generators of p-groups
• §16. Classification of finite p-groups all of whose noncyclic subgroups are normal
• §17. Counting theorems for regular p-groups
• §18. Counting theorems for irregular p-groups
• §19. Some additional counting theorems
• §20. Groups with small abelian subgroups and partitions
• §21. On the Schur multiplier and the commutator subgroup
• §22. On characters of p-groups
• §23. On subgroups of given exponent
• §24. Hall’s theorem on normal subgroups of given exponent
• §25. On the lattice of subgroups of a group
• §26. Powerful p-groups
• §27. p-groups with normal centralizers of all elements
• §28. p-groups with a uniqueness condition for nonnormal subgroups
• §29. On isoclinism
• §30. On p-groups with few nonabelian subgroups of order pp and exponent p
• §31. On p-groups with small p0-groups of operators
• §32. W. Gaschütz’s and P. Schmid’s theorems on p-automorphisms of p-groups
• §33. Groups of order pm with automorphisms of order pm-1, pm-2 or pm-3
• §34. Nilpotent groups of automorphisms
• §35. Maximal abelian subgroups of p-groups
• §36. Short proofs of some basic characterization theorems of finite p-group theory
• §37. MacWilliams’ theorem
• §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
• §40. On breadth and class number of p-groups
• §41. Groups in which every two noncyclic subgroups of the same order have the same rank
• §42. On intersections of some subgroups
• §43. On 2-groups with few cyclic subgroups of given order
• §44. Some characterizations of metacyclic p-groups
• §45. A counting theorem for p-groups of odd order
• Appendix 1. The Hall–Petrescu formula
• Appendix 2. Mann’s proof of monomiality of p-groups
• Appendix 3. Theorems of Isaacs on actions of groups
• Appendix 4. Freiman’s number-theoretical theorems
• Appendix 5. Another proof of Theorem 5.4
• Appendix 6. On the order of p-groups of given derived length
• Appendix 7. Relative indices of elements of p-groups
• Appendix 8. p-groups withabsolutely regular Frattini subgroup
• Appendix 9. On characteristic subgroups of metacyclic groups
• Appendix 10. On minimal characters of p-groups
• Appendix 11. On sums of degrees of irreducible characters
• Appendix 12. 2-groups whose maximal cyclic subgroups of order > 2 are self-centralizing
• Appendix 13. Normalizers of Sylow p-subgroups of symmetric groups
• Appendix 14. 2-groups with an involution contained in only one subgroup of order 4
• Appendix 15. A criterion for a group to be nilpotent
• Research problems and themes I
• Backmatter