##
**The theory of classes of groups. Translated from the 1997 Chinese original.**
*(English)*
Zbl 1005.20016

Mathematics and its Applications (Dordrecht). 505. Dordrecht: Kluwer Academic Publishers. Beijing: Science Press. xii, 258 p. (2000).

A set of groups is said to be a class of groups if this set contains a group \(G\), then it also contains all groups isomorphic to \(G\). Remarkable is the fact that classes of groups have played a subsidiary role in research of internal structure of non-simple finite groups at the primary stage. Further classes of groups have been investigated for themselves. As things turned out formations and Fitting classes were very useful for applications. Remind that a formation is a class of finite groups closed with respect to homomorphic images and finite subdirect products. Fitting classes are dual to formations. Remind that a class of finite groups \(\mathfrak F\) is called a Fitting class if it is closed under taking of subnormal subgroups and subnormal \(\mathfrak F\)-subgroups products. New results of formation theory and some results concerning Fitting classes are presented in the monograph.

Chapters 1 and 5 have a subsidiary character. Some general concepts of group theory and linear algebra playing an important role in the course of the book are considered there.

The second chapter is devoted to \(\mathfrak F\)-covering subgroups, \(\mathfrak F\)-projectors, \(\mathfrak F\)-injectors and \(\mathfrak F\)-normalizers. In particular it is proved the existence and conjugacy of \(\mathfrak F\)-injectors for groups such that the quotient group by the radical is \(\pi(\mathfrak F)\)-soluble.

Let \(\mathfrak X\) be a class of groups. The symbol \(N^{\mathfrak X}\) denotes the class of all groups such that normalizers of all Sylow subgroups belong to \(\mathfrak F\). In 1992, L. A. Shemetkov posed a problem in Gomel Algebraic Seminar. Problem (1) Find all local \(s\)-closed formations \(\mathfrak F\) such that \(N^{\mathfrak F}\subseteq{\mathfrak F}\) in the class \(\mathfrak S\). (2) Describe all local Shemetkov formations with the condition \(N^{\mathfrak F}\subseteq{\mathfrak F}\) in the class \(\mathfrak S\).

Remind that an \(s\)-closed formation \(\mathfrak F\) is called a Shemetkov formation if any minimal non-\(\mathfrak F\)-group (in the universe considered) is either a Shmidt group or has prime order. The first part of the problem is very extensive and difficult. A theorem proved in Chapter 3 gives a solution of the second part of the problem.

Theorem. Let \(\mathfrak F\) be a non-primary soluble \(s\)-closed local Shemetkov formation. Then \({\mathfrak S}\cap N^{\mathfrak F}\subseteq{\mathfrak F}\) if and only if \({\mathfrak F}\in\{{\mathfrak N}_\pi,{\mathfrak S}_\pi\}\) where \(\pi\) is any two-element subset of \(\pi({\mathfrak F})\).

In this theorem we have \({\mathfrak F}_\pi={\mathfrak S}_\pi\cap{\mathfrak F}\) where \({\mathfrak S}_\pi\) is the class of all soluble \(\pi\)-groups.

Besides in Chapter 3 the structure of a finite group with normalizers of Sylow subgroups having either odd indices or prime power indices is described.

The product \(\mathfrak{MH}\) of formations \(\mathfrak M\) and \(\mathfrak H\) is the class \((G\mid G^{\mathfrak H}\in{\mathfrak M})\) where \(G^{\mathfrak H}=\bigcap\{N\mid N\triangleleft G\) and \(G/N\in{\mathfrak H}\}\). In 1995, A. N. Skiba posed the following Problem in “The Kourovka notebook” [see The Kourovka notebook. Unsolved problems in group theory. Institute of Mathematics. Russian Academy of Sciences. Siberian Branch. Novosibirsk (1995; Zbl 0838.20001)]. Problem 12.74: If \({\mathfrak F}=\mathfrak{MH}\) is a one-generated composition formation where \(\mathfrak M\) and \(\mathfrak H\) are non-trivial formations, is \(\mathfrak M\) a composition formation?

In Chapter 4 it is constructed an example which gives a negative answer to this problem. Besides in Chapter 4 it is obtained a complete solution of V. A. Vedernikov’s problem [Vopr. Algebry 5, 28-34 (1990; Zbl 0742.20022)] of the structure of saturated formations with complemented subformations of type \({\mathfrak N}_p\).

In the monograph 29 open problems are raised. At the end of each chapter a special section is devoted to notes or supplementary information including the related historical background knowledge and problems that wait to be solved.

Chapters 1 and 5 have a subsidiary character. Some general concepts of group theory and linear algebra playing an important role in the course of the book are considered there.

The second chapter is devoted to \(\mathfrak F\)-covering subgroups, \(\mathfrak F\)-projectors, \(\mathfrak F\)-injectors and \(\mathfrak F\)-normalizers. In particular it is proved the existence and conjugacy of \(\mathfrak F\)-injectors for groups such that the quotient group by the radical is \(\pi(\mathfrak F)\)-soluble.

Let \(\mathfrak X\) be a class of groups. The symbol \(N^{\mathfrak X}\) denotes the class of all groups such that normalizers of all Sylow subgroups belong to \(\mathfrak F\). In 1992, L. A. Shemetkov posed a problem in Gomel Algebraic Seminar. Problem (1) Find all local \(s\)-closed formations \(\mathfrak F\) such that \(N^{\mathfrak F}\subseteq{\mathfrak F}\) in the class \(\mathfrak S\). (2) Describe all local Shemetkov formations with the condition \(N^{\mathfrak F}\subseteq{\mathfrak F}\) in the class \(\mathfrak S\).

Remind that an \(s\)-closed formation \(\mathfrak F\) is called a Shemetkov formation if any minimal non-\(\mathfrak F\)-group (in the universe considered) is either a Shmidt group or has prime order. The first part of the problem is very extensive and difficult. A theorem proved in Chapter 3 gives a solution of the second part of the problem.

Theorem. Let \(\mathfrak F\) be a non-primary soluble \(s\)-closed local Shemetkov formation. Then \({\mathfrak S}\cap N^{\mathfrak F}\subseteq{\mathfrak F}\) if and only if \({\mathfrak F}\in\{{\mathfrak N}_\pi,{\mathfrak S}_\pi\}\) where \(\pi\) is any two-element subset of \(\pi({\mathfrak F})\).

In this theorem we have \({\mathfrak F}_\pi={\mathfrak S}_\pi\cap{\mathfrak F}\) where \({\mathfrak S}_\pi\) is the class of all soluble \(\pi\)-groups.

Besides in Chapter 3 the structure of a finite group with normalizers of Sylow subgroups having either odd indices or prime power indices is described.

The product \(\mathfrak{MH}\) of formations \(\mathfrak M\) and \(\mathfrak H\) is the class \((G\mid G^{\mathfrak H}\in{\mathfrak M})\) where \(G^{\mathfrak H}=\bigcap\{N\mid N\triangleleft G\) and \(G/N\in{\mathfrak H}\}\). In 1995, A. N. Skiba posed the following Problem in “The Kourovka notebook” [see The Kourovka notebook. Unsolved problems in group theory. Institute of Mathematics. Russian Academy of Sciences. Siberian Branch. Novosibirsk (1995; Zbl 0838.20001)]. Problem 12.74: If \({\mathfrak F}=\mathfrak{MH}\) is a one-generated composition formation where \(\mathfrak M\) and \(\mathfrak H\) are non-trivial formations, is \(\mathfrak M\) a composition formation?

In Chapter 4 it is constructed an example which gives a negative answer to this problem. Besides in Chapter 4 it is obtained a complete solution of V. A. Vedernikov’s problem [Vopr. Algebry 5, 28-34 (1990; Zbl 0742.20022)] of the structure of saturated formations with complemented subformations of type \({\mathfrak N}_p\).

In the monograph 29 open problems are raised. At the end of each chapter a special section is devoted to notes or supplementary information including the related historical background knowledge and problems that wait to be solved.

Reviewer: N.T.Vorob’ev (Vitebsk)

### MSC:

20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |

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