p-Automorphisms of Finite p-Groups

by Evgenii I. Khukhro

• ISBN13: 9780521597173
• ISBN10: 052159717X
• Format: Paperback
• Copyright: 1998-02-13
• Publisher: Cambridge University Press

Summary

This book provides a detailed but concise account of the theory of structure of finite p-groups admitting p-automorphisms with few fixed points. The relevant preliminary material on Lie rings is introduced and the main theorems of the book on the solubility of finite p-groups are then presented. The proofs involve notions such as viewing automorphisms as linear transformations, associated Lie rings, powerful p-groups, and the correspondences of A.I. Mal'cev and M. Lazard given by the Baker-Hausdorff formula. Many exercises are included. This book is suitable for graduate students and researchers working in the fields of group theory and Lie rings.

Table of Contents

Preface	viii	(1)
Introduction	ix
Chapter 1. Preliminaries	1	(25)
1.1. Groups	1	(12)
1.2. Rings and modules	13	(4)
1.3. Algebraic systems, varieties and free objects	17	(7)
Exercises 1	24	(2)
Chapter 2. Automorphisms and their fixed points	26	(10)
2.1. Semidirect Products	26	(3)
2.2. Automorphisms as linear transformations	29	(3)
2.3. Induced automorphisms of factor-groups	32	(2)
Exercises 2	34	(2)
Chapter 3. Nilpotent and soluble groups	36	(15)
3.1. The lower central series	36	(2)
3.2. Nilpotent groups	38	(4)
3.3. Soluble groups and varieties	42	(4)
3.4. Nilpotency criteria for soluble groups	46	(2)
Exercises 3	48	(3)
Chapter 4. Finite p-groups	51	(6)
4.1. Basic properties	51	(3)
4.2. A theorem of P.Hall	54	(2)
Exercises 4	56	(1)
Chapter 5. Lie rings	57	(17)
5.1. Definitions and basic properties	57	(5)
5.2. Nilpotent and soluble Lie rings	62	(4)
5.3. Free Lie rings	66	(5)
Exercises 5	71	(3)
Chapter 6. Associated Lie rings	74	(9)
6.1. Definition	74	(3)
6.2. Basic properties	77	(2)
6.3. Some applications	79	(2)
Exercises 6	81	(2)
Chapter 7. Regular automorphisms of Lie rings	83	(12)
7.1. Graded Lie rings	83	(5)
7.2. Combinatorial consequences	88	(2)
7.3. Regular automorphisms	90	(3)
Exercises 7	93	(2)
Chapter 8. Almost regular automorphism of order p: almost nilpotency of p-bounded class	95	(6)
Exercises 8	99	(2)
Chapter 9. The Baker-Hausdorff Formula and nilpotent Q-powered groups	101	(12)
9.1. Free nilpotent groups	101	(3)
9.2. The Baker-Hausdorff Formula	104	(5)
9.3. Nilpotent Q-powered groups	109	(3)
Exercises 9	112	(1)
Chapter 10. The correspondences of A.I. Mal'cev and M. Lazard	113	(17)
10.1. The Mal'cev Correspondence	113	(8)
10.2. The Lazard Correspondence	121	(7)
Exercises 10	128	(2)
Chapter 11. Powerful p-groups	130	(10)
11.1. Definitions and basic properties	130	(5)
11.2. Finite p-groups of bounded rank	135	(3)
Exercises 11	138	(2)
Chapter 12. Almost regular automorphism of order p(n): almost solubility of p(n)-bounded derived length	140	(11)
12.1. Uniformly Powerful case	140	(3)
12.2. Application of the Mal'cev Correspondence	143	(3)
12.3. Almost solubility of p(n)-bounded derived length	146	(4)
Exercises 12	150	(1)
Chapter 13. p-Automorphisms with p fixed points	151	(15)
13.1. Abelian p-groups	152	(3)
13.2. Reduction to Lie rings	155	(5)
13.3. The Lie ring theorem	160	(5)
Exercises 13	165	(1)
Chapter 14. Automorphism of order p with p(m) fixed points: almost nilpotency of m-bounded class	166	(24)
14.1. Almost solubility of m-bounded derived length	167	(13)
14.2. Almost nilpotency of m-bounded class	180	(9)
Exercises 14	189	(1)
Bibliography	190	(8)
Index of names	198	(1)
Subject Index	199	(4)
List of symbols	203