##
**Group theory II. Transl. from the Japanese.**
*(English)*
Zbl 0586.20001

Grundlehren der Mathematischen Wissenschaften, 248. New York etc.: Springer-Verlag. x, 621 p. DM 268,00 (1986).

This is the second and final volume of the English translation of the Japanese original [Vol. I (1977; Zbl 0361.20001), English translation (1982; Zbl 0472.20001)]; Vol. II (1978; Zbl 0488.20002)]. This volume is devoted to finite groups and has surprisingly small overlaps with existing books.

Chapter 4. Commutators. §1. Commutator subgroups. Interesting feature of the author’s exposition of this standard theme are useful refinements of many well known theorems (see, for example Th. 1.6(ii), (1.12) and so on).

§2. Nilpotent groups. Corollary 3 from (2.10) on nilpotency of the stability group of a normal series due to L. A. Kaluzhnin (a generalization of this result to arbitrary series of subgroups, due to P. Hall, B. I. Plotkin and V. G. Viljačer, see ex. 6). It is proved that if the relation \(\rho =1\) holds in every finite \(p\)-group then \(\rho\) must be a trivial relation. A short discussion of Fitting and Frattini subgroups is given (in exercises are discussed Carter and Fischer subgroups).

§3. Commutator calculations. Identity of Hall-Petrescu. Basic properties of regular \(p\)-groups.

§4. Finite \(p\)-groups. Blackburn’s results on \(p\)-groups, \(p>2\), without subgroups of type \((p,p,p)\) (if \(G\) contains such a subgroup it contains as proved by this reviewer \(\equiv 1\pmod p\) such subgroups). We note that A. D. Ustuzhaninov in 1973 proved that a 2-group \(G\) without normal subgroups of type \((2,2,2)\) has a metacyclic normal subgroup \(M\) such that \(G/M\) is a subgroup of the dihedral group of order 8 (in particular \(G\) is generated by four elements). A short discussion of special and extra-special \(p\)-groups is given. (4.17) is proved by A. N. Fomin. (4.20) appeared in Ya. G. Berkovich [Sib. Mat. Zh. 9, 1284–1306 (1968; Zbl 0315.20015)]. (4.20) is due to C. Hobby [Pac. J. Math. 10, 209–212 (1960; Zbl 0090.01902)]. As to Hall’s theorem (4.22) on symplectic \(p\)-groups, its generalization see in Ya. G. Berkovich [Izv. Akad. Nauk SSSR, Ser. Mat. 35, 800–830 (1971; Zbl 0257.20014)] in which are studied \(p\)-groups with cyclic Frattini subgroup. Thompson-Glauberman replacement theorem. Higman’s estimate for the number \(f(n,p)\) of groups of order \(p^n\) (for \(n\le 6\) is it known to be the exact value for \(f(n,p))\). This section is finished by a brief discussion of isoclinism.

§5. Finite solvable groups. Elementary theory of solvable groups. Th. 5.12 on solvability of groups with Sylow system was proved independently by P. Hall and S. A. Chunikhin (1905–1985). Note that 5.6 may be proved without appealing to the Schur-Zassenhaus theorem.

Chapter 5. Finite groups I. §1. Elements of order two. The Brauer-Fowler paper (some fragments) on groups of even order. Bender’s papers on groups with large subgroup. Characterization (by centralizers) of \(A_5\), \(A_6\), \(A_7\) [in the book of Ya. G. Berkovich and È. M. Zhmud’, Characters of finite groups, I (to appear) in Ch. 15 is obtained by the same method a characterization of \(\mathrm{PSL}(2,2^n)\) (this characterization was first obtained in a forgotten 1900 paper of Burnside and rediscovered in the well-known Brauer-Suzuki-Wall paper (1958)].

§2. Transfer and fusion of elements. Theorem of D. G. Higman on focal subgroups. Result of J. L. Alperin on conjugation families. Conjugacy functor, section conjugacy functor. Tate’s theorem on normal p- complements (a character-theoretic proof, due to Thompson, see in the books of Dornhoff and Isaacs).

§3. Generalization of Sylow’s theorems. (P. Hall and S. A. Chunikhin results on \(D_{\pi}\) in \(\pi\)-separable groups (P. Hall’s main theorem D5 is given in an exercise). Theorem of S. A. Chunikhin on indexials. Some results of H. Wielandt on maximal \(\pi\)-subgroups.

§4. ZJ-theorem. Complete proof of Glauberman’s ZJ-theorem. Some of its consequences. Generalization of Thompson’s theorem on normal \(p\)-complements due to Glauberman. Thompson’s theorem on solvability of groups with a nilpotent maximal subgroup of odd order. Thompson’s theorem on nilpotency of the Frobenius kernel (this is the solution of a long-standing conjecture), Kegel’s theorem on nilpotency of \(H(p)\)-subgroups. Character-free proof of the \(p^aq^b\)-theorem of Burnside (due to Goldschmidt and Matsuyama).

§5. Supplements. Functors \(K^{\infty}\) and \(K_{\infty}\) and analogues of the ZJ-theorem for these functors. Non-simplicity of a group with self-normalizing Sylow p-subgroup, \(p\geq 5\) (G. Glauberman’s theorem), Yoshida’s character theoretic transfer (Isaacs’ exposition).

Chapter 6. Finite groups II. This long chapter (350 pages) is intended as an introduction to the recent progress in the theory of simple groups.

§1. Representation theory. Short essay without proof of the main results of representation theory. Basic concepts. Group rings. Irreducible representations. Induced representations. Characters. Generalized characters. Induction mappings. Central characters. Brauer characters.

§2. Applications of representation theory. Frobenius theorem (in the earlier mentioned book of Berkovich-Zhmud’ are proved many inversions of this theorem; for example, degrees of irreducible characters with multiplicities determine whether G is a Frobenius group or not). Exceptional characters. Suzuki’s theorem on the structure of CA-groups of odd order (see D. Gorenstein’s ”Finite groups” [New York etc.: Harper and Row Publ. (1968; Zbl 0185.05701)] on CN-groups of odd order). We note that L. Weisner determined the structure of solvable CA-groups. Even case of the CA-theorem (generalization of the early mentioned Brauer-Suzuki-Wall theorem). Brauer-Suzuki theorem on groups with generalized quaternion \(S_2\)-subgroup (this proof for \(16\nmid | G|\) is due to G. Glauberman). \(Z^*\)-theorem (a deep generalization preceding the Brauer-Suzuki theorem). Some results from modular representation theory. Hall-Higman theorem on matrix groups over fields of characteristic \(p>0\). Transitivity and uniqueness theorems for groups of odd order. Strongly embedded subgroups and some related results (this is a deep generalization of Suzuki’s theorem on groups with independent Sylow 2-subgroups).

§5. The classification problem of simple groups. A more complete discussion of these topics see in the survey articles and books of Gorenstein. Determination of the structure of centralizers of involutions in classical simple groups.

§6. The components of a group. Quasisimple and semisimple groups, components of a group, generalized Fitting subgroup, 2-components and 2- layer of a group.

§6. Signalizer functors. \(H<G\) is a signalizer if \(| G|\) and \(| G:H|\) are odd. If \(G\) is solvable then \(O(G)\) is the unique maximal signalizer of \(G\). The concepts of a signalizer functor arises from an attempt to generalize this property of solvable groups to a more general class of groups. Such an axiomatization is very useful and justified by recent work on simple groups. Goldschmidt’s theorem on completeness of signalizer functors. Glauberman’s result on solvable groups.

§8. Small simple groups. Brauer-Suzuki-Wall theorem on groups in which every two maximal cyclic subgroups of even order have trivial intersection. Automorphism groups of groups of Lie type. Some results (without proofs) on the determination of simple groups by their \(S_2\)-subgroups. In a supplement to the English edition the author reports on Bombieri’s results on groups of Ree type, on the complete classification of simple groups, on revisions of proofs of some fundamental theorems (on groups with dihedral \(S_2\)-subgroups, on groups of sectional rank at most 4 and so on).

§9. Two types of simple groups. Classification of simple groups of non-component type. Aschbacher’s results on groups with proper 2-generated core.

§10. Graphs and simple groups. The graphs associated with a transitive permutation group. Transitive permutation groups of rank three. Graphs and geometry. A theorem of Fischer. In a supplement to the English edition the author talks about new results on the Fischer-Griess simple group (Monster).

§11. Fusions in \(S_2\)-subgroups. Fusion theorem of Goldschmidt and certain applications.

§12. Recent developments in the theory of simple groups. Short survey.

This two-volume book is a good complement to the existing literature on finite groups. Many results and their proofs can be found only in this book. In spite of issuing Gorenstein’s survey articles and books on the classification problem we recommend to read this book before the mentioned material. In this year are issued two related books. J. H. Walter, The B-conjecture: characterization of Chevalley groups [Mem. Am. Math. Soc. 345 (1986)], and M. Aschbacher, Overgroups of Sylow subgroups in sporadic groups [ibid. 343 (1986; Zbl 0585.20005)].

Chapter 4. Commutators. §1. Commutator subgroups. Interesting feature of the author’s exposition of this standard theme are useful refinements of many well known theorems (see, for example Th. 1.6(ii), (1.12) and so on).

§2. Nilpotent groups. Corollary 3 from (2.10) on nilpotency of the stability group of a normal series due to L. A. Kaluzhnin (a generalization of this result to arbitrary series of subgroups, due to P. Hall, B. I. Plotkin and V. G. Viljačer, see ex. 6). It is proved that if the relation \(\rho =1\) holds in every finite \(p\)-group then \(\rho\) must be a trivial relation. A short discussion of Fitting and Frattini subgroups is given (in exercises are discussed Carter and Fischer subgroups).

§3. Commutator calculations. Identity of Hall-Petrescu. Basic properties of regular \(p\)-groups.

§4. Finite \(p\)-groups. Blackburn’s results on \(p\)-groups, \(p>2\), without subgroups of type \((p,p,p)\) (if \(G\) contains such a subgroup it contains as proved by this reviewer \(\equiv 1\pmod p\) such subgroups). We note that A. D. Ustuzhaninov in 1973 proved that a 2-group \(G\) without normal subgroups of type \((2,2,2)\) has a metacyclic normal subgroup \(M\) such that \(G/M\) is a subgroup of the dihedral group of order 8 (in particular \(G\) is generated by four elements). A short discussion of special and extra-special \(p\)-groups is given. (4.17) is proved by A. N. Fomin. (4.20) appeared in Ya. G. Berkovich [Sib. Mat. Zh. 9, 1284–1306 (1968; Zbl 0315.20015)]. (4.20) is due to C. Hobby [Pac. J. Math. 10, 209–212 (1960; Zbl 0090.01902)]. As to Hall’s theorem (4.22) on symplectic \(p\)-groups, its generalization see in Ya. G. Berkovich [Izv. Akad. Nauk SSSR, Ser. Mat. 35, 800–830 (1971; Zbl 0257.20014)] in which are studied \(p\)-groups with cyclic Frattini subgroup. Thompson-Glauberman replacement theorem. Higman’s estimate for the number \(f(n,p)\) of groups of order \(p^n\) (for \(n\le 6\) is it known to be the exact value for \(f(n,p))\). This section is finished by a brief discussion of isoclinism.

§5. Finite solvable groups. Elementary theory of solvable groups. Th. 5.12 on solvability of groups with Sylow system was proved independently by P. Hall and S. A. Chunikhin (1905–1985). Note that 5.6 may be proved without appealing to the Schur-Zassenhaus theorem.

Chapter 5. Finite groups I. §1. Elements of order two. The Brauer-Fowler paper (some fragments) on groups of even order. Bender’s papers on groups with large subgroup. Characterization (by centralizers) of \(A_5\), \(A_6\), \(A_7\) [in the book of Ya. G. Berkovich and È. M. Zhmud’, Characters of finite groups, I (to appear) in Ch. 15 is obtained by the same method a characterization of \(\mathrm{PSL}(2,2^n)\) (this characterization was first obtained in a forgotten 1900 paper of Burnside and rediscovered in the well-known Brauer-Suzuki-Wall paper (1958)].

§2. Transfer and fusion of elements. Theorem of D. G. Higman on focal subgroups. Result of J. L. Alperin on conjugation families. Conjugacy functor, section conjugacy functor. Tate’s theorem on normal p- complements (a character-theoretic proof, due to Thompson, see in the books of Dornhoff and Isaacs).

§3. Generalization of Sylow’s theorems. (P. Hall and S. A. Chunikhin results on \(D_{\pi}\) in \(\pi\)-separable groups (P. Hall’s main theorem D5 is given in an exercise). Theorem of S. A. Chunikhin on indexials. Some results of H. Wielandt on maximal \(\pi\)-subgroups.

§4. ZJ-theorem. Complete proof of Glauberman’s ZJ-theorem. Some of its consequences. Generalization of Thompson’s theorem on normal \(p\)-complements due to Glauberman. Thompson’s theorem on solvability of groups with a nilpotent maximal subgroup of odd order. Thompson’s theorem on nilpotency of the Frobenius kernel (this is the solution of a long-standing conjecture), Kegel’s theorem on nilpotency of \(H(p)\)-subgroups. Character-free proof of the \(p^aq^b\)-theorem of Burnside (due to Goldschmidt and Matsuyama).

§5. Supplements. Functors \(K^{\infty}\) and \(K_{\infty}\) and analogues of the ZJ-theorem for these functors. Non-simplicity of a group with self-normalizing Sylow p-subgroup, \(p\geq 5\) (G. Glauberman’s theorem), Yoshida’s character theoretic transfer (Isaacs’ exposition).

Chapter 6. Finite groups II. This long chapter (350 pages) is intended as an introduction to the recent progress in the theory of simple groups.

§1. Representation theory. Short essay without proof of the main results of representation theory. Basic concepts. Group rings. Irreducible representations. Induced representations. Characters. Generalized characters. Induction mappings. Central characters. Brauer characters.

§2. Applications of representation theory. Frobenius theorem (in the earlier mentioned book of Berkovich-Zhmud’ are proved many inversions of this theorem; for example, degrees of irreducible characters with multiplicities determine whether G is a Frobenius group or not). Exceptional characters. Suzuki’s theorem on the structure of CA-groups of odd order (see D. Gorenstein’s ”Finite groups” [New York etc.: Harper and Row Publ. (1968; Zbl 0185.05701)] on CN-groups of odd order). We note that L. Weisner determined the structure of solvable CA-groups. Even case of the CA-theorem (generalization of the early mentioned Brauer-Suzuki-Wall theorem). Brauer-Suzuki theorem on groups with generalized quaternion \(S_2\)-subgroup (this proof for \(16\nmid | G|\) is due to G. Glauberman). \(Z^*\)-theorem (a deep generalization preceding the Brauer-Suzuki theorem). Some results from modular representation theory. Hall-Higman theorem on matrix groups over fields of characteristic \(p>0\). Transitivity and uniqueness theorems for groups of odd order. Strongly embedded subgroups and some related results (this is a deep generalization of Suzuki’s theorem on groups with independent Sylow 2-subgroups).

§5. The classification problem of simple groups. A more complete discussion of these topics see in the survey articles and books of Gorenstein. Determination of the structure of centralizers of involutions in classical simple groups.

§6. The components of a group. Quasisimple and semisimple groups, components of a group, generalized Fitting subgroup, 2-components and 2- layer of a group.

§6. Signalizer functors. \(H<G\) is a signalizer if \(| G|\) and \(| G:H|\) are odd. If \(G\) is solvable then \(O(G)\) is the unique maximal signalizer of \(G\). The concepts of a signalizer functor arises from an attempt to generalize this property of solvable groups to a more general class of groups. Such an axiomatization is very useful and justified by recent work on simple groups. Goldschmidt’s theorem on completeness of signalizer functors. Glauberman’s result on solvable groups.

§8. Small simple groups. Brauer-Suzuki-Wall theorem on groups in which every two maximal cyclic subgroups of even order have trivial intersection. Automorphism groups of groups of Lie type. Some results (without proofs) on the determination of simple groups by their \(S_2\)-subgroups. In a supplement to the English edition the author reports on Bombieri’s results on groups of Ree type, on the complete classification of simple groups, on revisions of proofs of some fundamental theorems (on groups with dihedral \(S_2\)-subgroups, on groups of sectional rank at most 4 and so on).

§9. Two types of simple groups. Classification of simple groups of non-component type. Aschbacher’s results on groups with proper 2-generated core.

§10. Graphs and simple groups. The graphs associated with a transitive permutation group. Transitive permutation groups of rank three. Graphs and geometry. A theorem of Fischer. In a supplement to the English edition the author talks about new results on the Fischer-Griess simple group (Monster).

§11. Fusions in \(S_2\)-subgroups. Fusion theorem of Goldschmidt and certain applications.

§12. Recent developments in the theory of simple groups. Short survey.

This two-volume book is a good complement to the existing literature on finite groups. Many results and their proofs can be found only in this book. In spite of issuing Gorenstein’s survey articles and books on the classification problem we recommend to read this book before the mentioned material. In this year are issued two related books. J. H. Walter, The B-conjecture: characterization of Chevalley groups [Mem. Am. Math. Soc. 345 (1986)], and M. Aschbacher, Overgroups of Sylow subgroups in sporadic groups [ibid. 343 (1986; Zbl 0585.20005)].

Reviewer: Ya.G.Berkovich

### MSC:

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

20D05 | Finite simple groups and their classification |

20D06 | Simple groups: alternating groups and groups of Lie type |

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