Groups of Prime Power Order. / Volume 1 / / Yakov Berkovich.

• Superior document: Title is part of eBook package: De Gruyter DG Expositions in Mathematics Backlist eBook Package
• VerfasserIn: Berkovich, Yakov,
• Place / Publishing House: Berlin ;, Boston : : De Gruyter, , [2008]; ©2008
• Year of Publication: 2008
• Language: English
• Series: De Gruyter Expositions in Mathematics , 46
• Physical Description: 1 online resource (512 p.)
• Other title: 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
• Format: Mode of access: Internet via World Wide Web.
• ISBN: 9783110208221; 9783110494969; 9783110238570; 9783110238471; 9783110637205; 9783110212129; 9783110212136; 9783110209082
• ISSN: 0938-6572
• DOI: 10.1515/9783110208221
• Access: restricted access
• Hierarchical level: Monograph
• Statement of Responsibility: Yakov Berkovich.

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.