##
**Finite soluble groups.**
*(English)*
Zbl 0753.20001

De Gruyter Expositions in Mathematics. 4. Berlin etc.: W. de Gruyter. xiii, 891 p. (1992).

This impressive volume is aimed at giving an overall picture of the theory of finite soluble groups as it has developed during the past three decades. The authors’ hope that their book will serve as a basic reference in the subject area, as a text for postgraduate teaching and as a source of research ideas and techniques appears to rest on solid foundations. The book meets the specialist’s needs, being comprehensive by and large while concentrating on those parts of the subject where a coherent theoretical structure has emerged. Its accessibility to postgraduates is derived from its making available all the basic and prerequisite group- and representation theoretical tools in Chapters A and B, respectively, which take up more than 200 pages of the book. The reader is often referred to B. Huppert, Endliche Gruppen I [Springer-Verlag, Berlin-Heidelberg-New York (1967; Zbl 0217.07201)] and to B. Huppert and N. Blackburn, Finite groups II [Springer-Verlag, Berlin-Heidelberg-New York (1982; Zbl 0477.20001)] for proofs of results in the initial chapters. Results or their proofs not provided by these volumes are, as a rule, fully stated and proved in the present book. The wealth of instructive examples and exercises drawn quite frequently from the original research papers should also be attractive to postgraduate teaching.

Roger Carter’s discovery in 1961 [Math. Z. 75, 136-139 (1961; Zbl 0168.27301)] that each finite soluble group possesses a unique conjugacy class of self-normalizing nilpotent subgroups has been widely regarded as the start of the modern and post-modern development which has taken place in the theory of soluble groups. Only two major existence-and-conjugacy results had preceded the discovery of the Carter subgroups, namely Sylow’s theorems (1872), aptly dubbed by the authors the “Big Bang”, and Philip Hall’s theorems (1928).

Chapter I is mainly concerned with Hall’s results and techniques, comprising pronormality, Hall systems and their normalizers, and Hall’s renowned characterization of soluble groups by the existence of Sylow complements. The first two sections furnish an elementary purely group-theoretical proof of Burnside’s \(p^ aq^ b\)-theorem by the simplest methods, in an approach that has evolved from ideas of Feit and Thompson, Glauberman, Goldschmidt, Bender, Matsuyama among others.

W. Gaschütz’s seminal paper [Math. Z. 80, 300-305 (1963; Zbl 0111.244)] introduced saturated formations as the first families of group classes shown to breed canonically defined conjugacy classes of subgroups in every finite soluble group, called covering subgroups. Chapter II acquaints the reader with group classes in general, as objects of study per se, and closure properties defining them. Chapters III-VII deal with the “projective”, Chapters IX-XI with the “injective” portion of the theory.

Chapter III opens with a historical introduction, but the subsequent arrangement of the material is systematic rather than chronological. The account continues with Schunck classes and a most fruitful notion due to K. Doerk [J. Algebra 30, 12-30 (1974; Zbl 0346.20012)], the boundary of a Schunck class. P. Förster’s conception, starting with a paper of 1984 [Math. Z. 186, 149-178 (1984; Zbl 0544.20015)] which takes the theory beyond soluble groups, is adopted in clarifying the relationship between projectors and covering subgroups arising from a Schunck class.

Chapter IV is devoted to the theory of formations. Links between Schunck classes and formations are established. A section on local formations leads up to the theorem of U. Lubeseder [Formationsbildungen in endlichen auflösbaren Gruppen, Dissertation, Universität Kiel (1963)], which identifies them with the saturated formations. A hitherto unpublished generalization of this theorem, found by Baer, follows. The chapter concludes with results on projectors brought about by saturated formations and on groups of operators acting hypercentrally with respect to general formation functions.

Chapter V is centred upon the work by R. Carter and T. Hawkes [J. Algebra 5, 175-202 (1967; Zbl 0167.292)], extends Hall’s theory of system normalizers and thus makes it part and parcel of the theory of local formations. The final section, relying on an unpublished manuscript by B. Fischer, considers so-called precursive subgroups of which W. Gaschütz’s prefrattini subgroups [Arch. Math. 13, 418-426 (1962; Zbl 0109.01403)] and the normalizers relative to a local formation are examples.

Chapter VI offers additional topics in Schunck class theory: the lattice structure of the family of Schunck classes with respect to the relation “being strongly contained in”; \(D\)-classes in the sense of G. Wood [Math. Z. 130, 31-37 (1973; Zbl 0239.20016)]; work by P. Förster [J. Algebra 55, 155-187 (1978; Zbl 0373.20015); Math. Z. 162, 219-234 (1978; Zbl 0373.20016)] on Schunck classes for which the projectors they define have special properties.

Chapter VII starts off with the proof of the finiteness theorem by R. Bryant, R. Bryce and B. Hartley [Bull. Aust. Math. Soc. 2, 347- 357 (1970; Zbl 0191.022)] on formations and saturated formations generated by a single soluble group. Section 2 focuses on the role of chief factor rank and culminates in the theorem of Harman [Characterizations of some classes of finite groups, Ph. D. thesis, University of Warwick (1981)] and H. Heineken [Boll. Unione Mat. Ital., V. Ser. B 16, 754-764 (1979; Zbl 0409.20014)]. Sections 3-5 penetrate further into the theory of saturated formations: characterization of primitive saturated formations by various closure properties, the theorem of Cossey and Oates-Macdonald on the saturated formation generated by a class of groups, K. Doerk’s work [Arch. Math. 28, 561-571 (1977; Zbl 0403.20011)] on the saturation of a formation, strong containment for saturated formations, the theory of extreme classes and \(s\)-critical groups by R. Carter, B. Fischer and T. Hawkes [J. Algebra 9, 285-315 (1968; Zbl 0177.039)] and saturated formations with the cover-avoidance property.

Chapter VIII is the first chapter to go into injective classes. H. Fitting’s achievement [Jber. Deutsch. Math.-Verein. 48, 77-141 (1938; Zbl 0019.19801)] in having proved that nilpotent groups are normal-product closed can be seen as historically isolated from the evolution of a later period. The short and elegant paper by B. Fischer, W. Gaschütz and B. Hartley [Math. Z. 102, 337-339 (1967; Zbl 0183.029)] marks the onset of sustained research into injective classes. Injective classes are brought in by the authors via Fitting sets, which admit greater flexibility.

The systematic treatment of Fitting classes begins at Chapter IX with an introductory section. We encounter several constructions of Fitting classes, particular types of Fitting classes, e.g. Fischer classes, some characterizations of injectors, the extremely useful R. Dark construction [Math. Z. 127, 145-156 (1972; Zbl 0226.20013)] and variations.

Chapter X examines the important partition of the family of Fitting classes into the F. Lockett sections [Math. Z. 137, 131-136 (1974; Zbl 0286.20017)], the upper-star and lower-star operations and their connection with the relation “being normal in” between Fitting classes, which goes back to D. Blessenohl and W. Gaschütz [Math. Z. 118, 1-8 (1970; Zbl 0208.033)]. The significance of the wreath product for the general theory of Fitting classes is covered by a section based on work of P. Hauck [Zur Theorie der Fittingklassen endlicher auflösbarer Gruppen, Dissertation, Universität Mainz (1977), J. Algebra 59, 313-329 (1979; Zbl 0425.20018)]. Another section gives consideration to the H. Lausch group [Math. Z. 130, 67-72 (1973; Zbl 0238.20039)] of a class of finite groups as a tool for handling Lockett sections. It is used in section 5, which presents T. Berger’s theorem [Proc. Lond. Math. Soc., III. Ser. 42, 59-86 (1981; Zbl 0456.20006)] with the proof by O. Brison [J. Algebra 68, 31-53 (1981; Zbl 0455.20015)]. The theorem entails algorithms for the evaluation of radicals defined by certain lower-star Fitting classes (such as the lower star of the smallest normal Fitting class). A discussion of the Lockett conjecture rounds the chapter off.

Chapter XI contains the remarkable theorem of R. Bryce and J. Cossey [Math. Proc. Camb. Philos. Soc. 91, 225-258 and 343 (1982; Zbl 0487.20011)], which states that a subgroup-closed Fitting class of finite groups is a saturated formation, and work by the same authors [J. Aust. Math. Soc. 17, 285-304 (1974; Zbl 0292.20016)], including their joint paper with T. Berger [J. Aust. Math. Soc., Ser. A 38, 157-163 (1985; Zbl 0558.20015)] on metanilpotent Fitting classes; a different way of viewing these classes by K. Johnsen and H. Laue [Arch. Math. 30, 350-360 (1978; Zbl 0363.20014)]; Fitting class boundaries; and Frattini duals.

Two appendices, “A theorem of Oates and Powell” and “Frattini extensions”, a bibliography and two indices are at the end of the book.

The production is excellent. The authors prove: pleasant style is compatible with precision. The book has been awaited for many years. Now, as the book is out, the prediction may not be too bold that it will be a standard reference and a strong stimulus for new research in many years to come.

Roger Carter’s discovery in 1961 [Math. Z. 75, 136-139 (1961; Zbl 0168.27301)] that each finite soluble group possesses a unique conjugacy class of self-normalizing nilpotent subgroups has been widely regarded as the start of the modern and post-modern development which has taken place in the theory of soluble groups. Only two major existence-and-conjugacy results had preceded the discovery of the Carter subgroups, namely Sylow’s theorems (1872), aptly dubbed by the authors the “Big Bang”, and Philip Hall’s theorems (1928).

Chapter I is mainly concerned with Hall’s results and techniques, comprising pronormality, Hall systems and their normalizers, and Hall’s renowned characterization of soluble groups by the existence of Sylow complements. The first two sections furnish an elementary purely group-theoretical proof of Burnside’s \(p^ aq^ b\)-theorem by the simplest methods, in an approach that has evolved from ideas of Feit and Thompson, Glauberman, Goldschmidt, Bender, Matsuyama among others.

W. Gaschütz’s seminal paper [Math. Z. 80, 300-305 (1963; Zbl 0111.244)] introduced saturated formations as the first families of group classes shown to breed canonically defined conjugacy classes of subgroups in every finite soluble group, called covering subgroups. Chapter II acquaints the reader with group classes in general, as objects of study per se, and closure properties defining them. Chapters III-VII deal with the “projective”, Chapters IX-XI with the “injective” portion of the theory.

Chapter III opens with a historical introduction, but the subsequent arrangement of the material is systematic rather than chronological. The account continues with Schunck classes and a most fruitful notion due to K. Doerk [J. Algebra 30, 12-30 (1974; Zbl 0346.20012)], the boundary of a Schunck class. P. Förster’s conception, starting with a paper of 1984 [Math. Z. 186, 149-178 (1984; Zbl 0544.20015)] which takes the theory beyond soluble groups, is adopted in clarifying the relationship between projectors and covering subgroups arising from a Schunck class.

Chapter IV is devoted to the theory of formations. Links between Schunck classes and formations are established. A section on local formations leads up to the theorem of U. Lubeseder [Formationsbildungen in endlichen auflösbaren Gruppen, Dissertation, Universität Kiel (1963)], which identifies them with the saturated formations. A hitherto unpublished generalization of this theorem, found by Baer, follows. The chapter concludes with results on projectors brought about by saturated formations and on groups of operators acting hypercentrally with respect to general formation functions.

Chapter V is centred upon the work by R. Carter and T. Hawkes [J. Algebra 5, 175-202 (1967; Zbl 0167.292)], extends Hall’s theory of system normalizers and thus makes it part and parcel of the theory of local formations. The final section, relying on an unpublished manuscript by B. Fischer, considers so-called precursive subgroups of which W. Gaschütz’s prefrattini subgroups [Arch. Math. 13, 418-426 (1962; Zbl 0109.01403)] and the normalizers relative to a local formation are examples.

Chapter VI offers additional topics in Schunck class theory: the lattice structure of the family of Schunck classes with respect to the relation “being strongly contained in”; \(D\)-classes in the sense of G. Wood [Math. Z. 130, 31-37 (1973; Zbl 0239.20016)]; work by P. Förster [J. Algebra 55, 155-187 (1978; Zbl 0373.20015); Math. Z. 162, 219-234 (1978; Zbl 0373.20016)] on Schunck classes for which the projectors they define have special properties.

Chapter VII starts off with the proof of the finiteness theorem by R. Bryant, R. Bryce and B. Hartley [Bull. Aust. Math. Soc. 2, 347- 357 (1970; Zbl 0191.022)] on formations and saturated formations generated by a single soluble group. Section 2 focuses on the role of chief factor rank and culminates in the theorem of Harman [Characterizations of some classes of finite groups, Ph. D. thesis, University of Warwick (1981)] and H. Heineken [Boll. Unione Mat. Ital., V. Ser. B 16, 754-764 (1979; Zbl 0409.20014)]. Sections 3-5 penetrate further into the theory of saturated formations: characterization of primitive saturated formations by various closure properties, the theorem of Cossey and Oates-Macdonald on the saturated formation generated by a class of groups, K. Doerk’s work [Arch. Math. 28, 561-571 (1977; Zbl 0403.20011)] on the saturation of a formation, strong containment for saturated formations, the theory of extreme classes and \(s\)-critical groups by R. Carter, B. Fischer and T. Hawkes [J. Algebra 9, 285-315 (1968; Zbl 0177.039)] and saturated formations with the cover-avoidance property.

Chapter VIII is the first chapter to go into injective classes. H. Fitting’s achievement [Jber. Deutsch. Math.-Verein. 48, 77-141 (1938; Zbl 0019.19801)] in having proved that nilpotent groups are normal-product closed can be seen as historically isolated from the evolution of a later period. The short and elegant paper by B. Fischer, W. Gaschütz and B. Hartley [Math. Z. 102, 337-339 (1967; Zbl 0183.029)] marks the onset of sustained research into injective classes. Injective classes are brought in by the authors via Fitting sets, which admit greater flexibility.

The systematic treatment of Fitting classes begins at Chapter IX with an introductory section. We encounter several constructions of Fitting classes, particular types of Fitting classes, e.g. Fischer classes, some characterizations of injectors, the extremely useful R. Dark construction [Math. Z. 127, 145-156 (1972; Zbl 0226.20013)] and variations.

Chapter X examines the important partition of the family of Fitting classes into the F. Lockett sections [Math. Z. 137, 131-136 (1974; Zbl 0286.20017)], the upper-star and lower-star operations and their connection with the relation “being normal in” between Fitting classes, which goes back to D. Blessenohl and W. Gaschütz [Math. Z. 118, 1-8 (1970; Zbl 0208.033)]. The significance of the wreath product for the general theory of Fitting classes is covered by a section based on work of P. Hauck [Zur Theorie der Fittingklassen endlicher auflösbarer Gruppen, Dissertation, Universität Mainz (1977), J. Algebra 59, 313-329 (1979; Zbl 0425.20018)]. Another section gives consideration to the H. Lausch group [Math. Z. 130, 67-72 (1973; Zbl 0238.20039)] of a class of finite groups as a tool for handling Lockett sections. It is used in section 5, which presents T. Berger’s theorem [Proc. Lond. Math. Soc., III. Ser. 42, 59-86 (1981; Zbl 0456.20006)] with the proof by O. Brison [J. Algebra 68, 31-53 (1981; Zbl 0455.20015)]. The theorem entails algorithms for the evaluation of radicals defined by certain lower-star Fitting classes (such as the lower star of the smallest normal Fitting class). A discussion of the Lockett conjecture rounds the chapter off.

Chapter XI contains the remarkable theorem of R. Bryce and J. Cossey [Math. Proc. Camb. Philos. Soc. 91, 225-258 and 343 (1982; Zbl 0487.20011)], which states that a subgroup-closed Fitting class of finite groups is a saturated formation, and work by the same authors [J. Aust. Math. Soc. 17, 285-304 (1974; Zbl 0292.20016)], including their joint paper with T. Berger [J. Aust. Math. Soc., Ser. A 38, 157-163 (1985; Zbl 0558.20015)] on metanilpotent Fitting classes; a different way of viewing these classes by K. Johnsen and H. Laue [Arch. Math. 30, 350-360 (1978; Zbl 0363.20014)]; Fitting class boundaries; and Frattini duals.

Two appendices, “A theorem of Oates and Powell” and “Frattini extensions”, a bibliography and two indices are at the end of the book.

The production is excellent. The authors prove: pleasant style is compatible with precision. The book has been awaited for many years. Now, as the book is out, the prediction may not be too bold that it will be a standard reference and a strong stimulus for new research in many years to come.

Reviewer: Hans Lausch (Clayton)

### MSC:

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

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

20F17 | Formations of groups, Fitting classes |

20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |