\(p\)-automorphisms of finite \(p\)-groups.

*(English)*Zbl 0897.20018
London Mathematical Society Lecture Note Series. 246. Cambridge: Cambridge University Press. xvii, 204 p. (1998).

The main object of the present book is a finite \(p\)-group admitting an automorphism of order \(p^k\) with exactly \(p^m\) fixed points. In recent years considerable progress has been made in the investigation of such groups. Various new results and methods have been introduced. Many of them have been obtained by the author. The book gives a complete, detailed account of new and old theorems in the area. Another purpose is presenting linear methods in the theory of the nilpotent groups: application of the associated Lie rings, theory of powerful \(p\)-groups, the correspondences of A. I. Mal’cev and M. Lazard, based on the Baker-Hausdorff formula. The book is self-contained, the proofs are very detailed and accessible for readers with a basic knowledge of group theory.

Chapters 1-4 contain preliminary material on groups, rings, modules, automorphisms, starting from the elementary principles. Lie rings and the associated Lie rings are introduced in chapters 5 and 6.

Chapter 7 is devoted to Lie rings with regular (without non-trivial fixed points) automorphisms. The classical results in the area are due to G. Higman, V. A. Kreknin and A. I. Kostrikin: (i) If a Lie ring \(L\) admits a regular automorphism of prime order \(p\) then \(L\) is nilpotent, of nilpotency class bounded by a function of \(p\) (G. Higman, 1957). In 1963 V. A. Kreknin and A. I. Kostrikin found a new proof of this theorem. (ii) If a Lie ring \(L\) admits a regular automorphism of arbitrary finite order \(n\) then \(L\) is soluble, of derived length bounded by a function of \(n\) (V. A. Kreknin, 1963).

Although a \(p\)-automorphism of a finite \(p\)-group can never be regular, combinatorial forms of the above theorems turned out to be useful for the investigation of finite \(p\)-groups with \(p\)-automorphisms. They are used in the proofs of all main results in the book.

Chapter 8 presents the first theorem concerning \(p\)-automorphisms of \(p\)-groups: If a finite \(p\)-group \(P\) admits an automorphism \(\varphi\) of prime order \(p\) with exactly \(p^m\) fixed points then \(P\) contains a subgroup of \((p,m)\)-bounded index which is nilpotent of \(p\)-bounded class (Alperin 1962, E. I. Khukhro 1985). The proof relies on the application of the associated Lie rings and Higman’s theorem.

Chapters 9-11 continue to introduce linear methods in the theory of nilpotent groups. The Baker-Hausdorff formula proved in chapter 9 is used in chapter 10 for stating the Mal’cev correspondence (between nilpotent \(Q\)-powered groups and nilpotent Lie \(Q\)-algebras) and the Lazard correspondence (for nilpotent \(p\)-groups of class at most \(p-1\)). The subject of chapter 11 is the theory of powerful \(p\)-groups recently developed by A. Lubotzky and A. Mann.

The second of the main results on \(p\)-automorphisms of \(p\)-groups is proved in chapter 12: If a finite \(p\)-group \(P\) admits an automorphism \(\varphi\) of order \(p^n\) with exactly \(p^m\) fixed points then \(P\) contains a subgroup of \((p,n,m)\)-bounded index which is nilpotent of \(p^n\)-bounded class (A. Shalev 1993, E. I. Khukhro 1993). The proof uses techniques accumulated in the previous chapters: Mal’cev correspondence, theory of powerful groups, Kreknin’s theorem.

The two final chapters 13 and 14 have particular interest because of remarkable methods applied in the proofs. They contain two mains theorems: (i) If a finite \(p\)-group \(P\) admits an automorphism \(\varphi\) of order \(p^n\) with only \(p\) fixed points then \(P\) contains a subgroup of \((p,n)\)-bounded index which is nilpotent of class at most \(2\) (for \(|\varphi|=p\), R. Shepherd, 1971, and C. R. Leedham-Green and S. McKay, 1976; for \(|\varphi|=p^n\), S. McKay, 1987, and I. Kiming, 1988). (ii) If a finite \(p\)-group \(P\) admits an automorphism \(\varphi\) of prime order \(p\) with exactly \(p^m\) fixed points then \(P\) contains a subgroup of \((p,m)\)-bounded index which is nilpotent of \(m\)-bounded class (Yu. A. Medvedev, 1994).

The proof of (i) is different from the original one and based on the same ideas as the proof of (ii). By application of the various linear methods from previous chapters (Higman’s and Kreknin’s theorems, powerful \(p\)-groups, Lazard correspondence) the theorems are reduced to the analogous theorems on Lie rings, which have an independent interest. The main idea in the proof of Lie ring theorems is to define a new ‘lifted’ Lie ring multiplication invented by Yu. Medvedev. This construction was anticipated by A. Shalev and E. I. Zel’manov in the works on \(p\)-groups and pro-\(p\)-groups of given coclass.

Each chapter is provided with exercises of various difficulty from elementary ones to results from research papers. The remarks of the later chapters point out some open problems in the area.

Chapters 1-4 contain preliminary material on groups, rings, modules, automorphisms, starting from the elementary principles. Lie rings and the associated Lie rings are introduced in chapters 5 and 6.

Chapter 7 is devoted to Lie rings with regular (without non-trivial fixed points) automorphisms. The classical results in the area are due to G. Higman, V. A. Kreknin and A. I. Kostrikin: (i) If a Lie ring \(L\) admits a regular automorphism of prime order \(p\) then \(L\) is nilpotent, of nilpotency class bounded by a function of \(p\) (G. Higman, 1957). In 1963 V. A. Kreknin and A. I. Kostrikin found a new proof of this theorem. (ii) If a Lie ring \(L\) admits a regular automorphism of arbitrary finite order \(n\) then \(L\) is soluble, of derived length bounded by a function of \(n\) (V. A. Kreknin, 1963).

Although a \(p\)-automorphism of a finite \(p\)-group can never be regular, combinatorial forms of the above theorems turned out to be useful for the investigation of finite \(p\)-groups with \(p\)-automorphisms. They are used in the proofs of all main results in the book.

Chapter 8 presents the first theorem concerning \(p\)-automorphisms of \(p\)-groups: If a finite \(p\)-group \(P\) admits an automorphism \(\varphi\) of prime order \(p\) with exactly \(p^m\) fixed points then \(P\) contains a subgroup of \((p,m)\)-bounded index which is nilpotent of \(p\)-bounded class (Alperin 1962, E. I. Khukhro 1985). The proof relies on the application of the associated Lie rings and Higman’s theorem.

Chapters 9-11 continue to introduce linear methods in the theory of nilpotent groups. The Baker-Hausdorff formula proved in chapter 9 is used in chapter 10 for stating the Mal’cev correspondence (between nilpotent \(Q\)-powered groups and nilpotent Lie \(Q\)-algebras) and the Lazard correspondence (for nilpotent \(p\)-groups of class at most \(p-1\)). The subject of chapter 11 is the theory of powerful \(p\)-groups recently developed by A. Lubotzky and A. Mann.

The second of the main results on \(p\)-automorphisms of \(p\)-groups is proved in chapter 12: If a finite \(p\)-group \(P\) admits an automorphism \(\varphi\) of order \(p^n\) with exactly \(p^m\) fixed points then \(P\) contains a subgroup of \((p,n,m)\)-bounded index which is nilpotent of \(p^n\)-bounded class (A. Shalev 1993, E. I. Khukhro 1993). The proof uses techniques accumulated in the previous chapters: Mal’cev correspondence, theory of powerful groups, Kreknin’s theorem.

The two final chapters 13 and 14 have particular interest because of remarkable methods applied in the proofs. They contain two mains theorems: (i) If a finite \(p\)-group \(P\) admits an automorphism \(\varphi\) of order \(p^n\) with only \(p\) fixed points then \(P\) contains a subgroup of \((p,n)\)-bounded index which is nilpotent of class at most \(2\) (for \(|\varphi|=p\), R. Shepherd, 1971, and C. R. Leedham-Green and S. McKay, 1976; for \(|\varphi|=p^n\), S. McKay, 1987, and I. Kiming, 1988). (ii) If a finite \(p\)-group \(P\) admits an automorphism \(\varphi\) of prime order \(p\) with exactly \(p^m\) fixed points then \(P\) contains a subgroup of \((p,m)\)-bounded index which is nilpotent of \(m\)-bounded class (Yu. A. Medvedev, 1994).

The proof of (i) is different from the original one and based on the same ideas as the proof of (ii). By application of the various linear methods from previous chapters (Higman’s and Kreknin’s theorems, powerful \(p\)-groups, Lazard correspondence) the theorems are reduced to the analogous theorems on Lie rings, which have an independent interest. The main idea in the proof of Lie ring theorems is to define a new ‘lifted’ Lie ring multiplication invented by Yu. Medvedev. This construction was anticipated by A. Shalev and E. I. Zel’manov in the works on \(p\)-groups and pro-\(p\)-groups of given coclass.

Each chapter is provided with exercises of various difficulty from elementary ones to results from research papers. The remarks of the later chapters point out some open problems in the area.

Reviewer: N.Yu.Makarenko (Novosibirsk)

##### MSC:

20D45 | Automorphisms of abstract finite groups |

20D15 | Finite nilpotent groups, \(p\)-groups |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20F40 | Associated Lie structures for groups |

20F18 | Nilpotent groups |

20F12 | Commutator calculus |

20F28 | Automorphism groups of groups |