Groups of prime power order. Volume 1 / by Yakov Berkovich.

Format:

Book

Author/Creator:

Berkovich, Yakov.

Series:

Gruyter expositions in mathematics ; 46.

De Gruyter expositions in mathematics, 0938-6572 ; 46

Language:

English

Subjects (All):

Finite groups.

Group theory.

Physical Description:

1 online resource (532 p.)

Edition:

1st ed.

Place of Publication:

Berlin ; New York : W. de Gruyter, c2008.

Language Note:

English

Summary:

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.

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

Notes:

Description based upon print version of record.

Includes bibliographical references and indexes.

ISBN:

9786611993474

9781281993472

1281993476

9783110208221

3110208229

OCLC:

808801315