##
**Groups and characters.**
*(English)*
Zbl 0896.20001

Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs and Tracts. New York, NY: John Wiley & Sons (ISBN 0-471-16340-6/hbk). viii, 212 p. (1997).

Contents: 1. Preliminaries. 2. Some groups. 3. Counting with groups. 4. Transfer and splitting. 5. Representations and characters. 6. Induction and restriction. 7. Computing character tables. 8. Characters of \(S_n\) and \(A_n\). 9. Frobenius groups. 10. Splitting fields.

This graduate-level text consists of two parts. Part I (Chapters 1-4) covers some basic aspects of finite group theory.

In Section 1.1 a very short proof of Schreier’s Theorem on generation of subgroups is given. As illustration, two Schreier generators for \(A_n\) are presented. In Section 1.2 the Schreier-Sims algorithm is described and explained in detail. As an easy consequence, it is shown that any subgroup of \(S_n\) is generated by \(n(n-1)/2\) elements (according to a deeper result of P. Neumann, if \(n>3\), every subgroup of \(S_n\) is generated by \([n/2]\) elements). In Section 1.3 the Coxeter-Todd algorithm is explained and illustrated. In Sections 1.4 and 1.5 semidirect products and wreath products are determined. In Section 1.6 transitivity and primitivity are studied. The Iwasawa criterion for a primitive group to be simple is proved. In Section 1.7 bilinear forms are considered. One proves that if \(B\) is a bilinear form on a vector space \(V\) such that \(B(u,v)B(w,u)=B(v,u)B(u,w)\) for all \(u,v,w\in V\) then \(B\) is symmetric or alternate.

In Chapter 2 some example of groups are studied in fairly complete detail. In Section 2.1 the automorphism group of a cyclic group of order \(n\) is described. In Section 2.2 the metacyclic groups are introduced. The decomposition of a Sylow \(p\)-subgroup of \(S_n\) in a direct product of indecomposable factors is given. In Section 2.3 finite subgroups of \(O(\mathbb{R}^2)\) and \(O(\mathbb{R}^3)\) are classified (Leonardo da Vinci and J. F. C. Hessel, respectively). In Section 2.4 the groups \(\text{GL}(n,F)\), \(\text{SL}(n,F)\) and \(\text{PSL}(n,F)\) are studied. One proves that \(\text{SL}(n,F)\) is generated by transvections, that \(\text{PSL}(n,F)\) is simple (with few exceptions). In Section 2.5 the five Mathieu groups are defined and some of their properties are proved (simplicity, orders, some subgroups, and so on). The symplectic groups \(\text{Sp}(V)\) are subject of Section 2.6. Using Iwasawa’s Theorem, one proves that \(\text{PSp}(V)\) is simple (with few exceptions).

The main topic of Chapter 3 is the Readfield-Polya Theorem that makes it possible to count orbits (it is a deep refinement of the Cauchy-Frobenius-Burnside Orbit Formula). Other topics: generating functions and Petersen graph.

In Chapter 4, using transfer, the focal subgroup theorem is proved. Next, the theorem on the structure of groups all whose Sylow subgroups are abelian, is proved. In Section 4.2 the Schur-Zassenhaus Theorem is proved. As a consequence, the Hall Theorem on solvable groups is proved (it is known that there is a proof of Hall’s Theorem independent of Schur-Zassenhaus’). In Section 4.3 some elementary results on \(p\)-groups are proved.

In Section 5.1 some basic results of representation theory are proved (Maschke’s Theorem, Schur’s Lemma, Schur’s relations for matrix elements of irreducible representations). In Section 5.2 one proves the first and second orthogonality relations and some of their consequences, and also Burnside’s \(p^aq^b\)-Theorem. In Section 5.3 irreducible characters of direct products are described. The Burnside-Brauer Theorem on powers of a faithful character is presented.

In Chapter 6 some standard results concentrating around Frobenius reciprocity and Clifford’s Theorem, are proved. Itô’s Theorem (\(\chi(1)\) divides \(| G:A|\) for an abelian normal subgroup \(A\) and \(\chi\in\text{Irr}(G)\)) is presented. One proves that nilpotent groups are monomial. In Section 6.4 some Mackey theorems are proved. A few character tables of small groups are constructed. In Section 6.5 Brauer’s induction theorems are proved (the proofs presented are due to Goldschmidt-Isaacs).

In Chapter 7, some algorithms for calculating the character table of a finite group \(G\) are discussed. The first algorithm, that is due to Burnside, is based on the fact that the structure constants of the class algebra of \(G\) determine the character table of \(G\). J. Dixon produced a revised form of Burnside’s algorithm (Section 7.2). G. J. A. Schneider discovered a variation on the algorithm that in many cases speeds it up considerably (Section 7.3).

Chapter 8 is an introduction to the Young theory of representations of \(S_n\).

Chapter 9 is devoted to Frobenius groups. All material is standard. An example of a Frobenius group with nonabelian kernel that is due to N. Itô, is presented (the first such example of order \(7^3\) was given by Otto Ju. Shmidt).

In Chapter 10 some advanced topics are treated. Brauer’s Realization Theorem is proved in Section 10.1. Section 10.2 is an introduction to the Schur index. The Frobenius-Schur indicator is a subject of Section 10.3; some details are given for Frobenius groups.

The book is well written. It contains some fresh material, esp. in Part 1. Some books with related theme that designate to the same audience, are T. Tambour, Introduction to finite groups and their representations [Lund (1991)] and J. L. Alperin and R. B. Bell, Groups and representations [Springer (1995; Zbl 0839.20001)].

This graduate-level text consists of two parts. Part I (Chapters 1-4) covers some basic aspects of finite group theory.

In Section 1.1 a very short proof of Schreier’s Theorem on generation of subgroups is given. As illustration, two Schreier generators for \(A_n\) are presented. In Section 1.2 the Schreier-Sims algorithm is described and explained in detail. As an easy consequence, it is shown that any subgroup of \(S_n\) is generated by \(n(n-1)/2\) elements (according to a deeper result of P. Neumann, if \(n>3\), every subgroup of \(S_n\) is generated by \([n/2]\) elements). In Section 1.3 the Coxeter-Todd algorithm is explained and illustrated. In Sections 1.4 and 1.5 semidirect products and wreath products are determined. In Section 1.6 transitivity and primitivity are studied. The Iwasawa criterion for a primitive group to be simple is proved. In Section 1.7 bilinear forms are considered. One proves that if \(B\) is a bilinear form on a vector space \(V\) such that \(B(u,v)B(w,u)=B(v,u)B(u,w)\) for all \(u,v,w\in V\) then \(B\) is symmetric or alternate.

In Chapter 2 some example of groups are studied in fairly complete detail. In Section 2.1 the automorphism group of a cyclic group of order \(n\) is described. In Section 2.2 the metacyclic groups are introduced. The decomposition of a Sylow \(p\)-subgroup of \(S_n\) in a direct product of indecomposable factors is given. In Section 2.3 finite subgroups of \(O(\mathbb{R}^2)\) and \(O(\mathbb{R}^3)\) are classified (Leonardo da Vinci and J. F. C. Hessel, respectively). In Section 2.4 the groups \(\text{GL}(n,F)\), \(\text{SL}(n,F)\) and \(\text{PSL}(n,F)\) are studied. One proves that \(\text{SL}(n,F)\) is generated by transvections, that \(\text{PSL}(n,F)\) is simple (with few exceptions). In Section 2.5 the five Mathieu groups are defined and some of their properties are proved (simplicity, orders, some subgroups, and so on). The symplectic groups \(\text{Sp}(V)\) are subject of Section 2.6. Using Iwasawa’s Theorem, one proves that \(\text{PSp}(V)\) is simple (with few exceptions).

The main topic of Chapter 3 is the Readfield-Polya Theorem that makes it possible to count orbits (it is a deep refinement of the Cauchy-Frobenius-Burnside Orbit Formula). Other topics: generating functions and Petersen graph.

In Chapter 4, using transfer, the focal subgroup theorem is proved. Next, the theorem on the structure of groups all whose Sylow subgroups are abelian, is proved. In Section 4.2 the Schur-Zassenhaus Theorem is proved. As a consequence, the Hall Theorem on solvable groups is proved (it is known that there is a proof of Hall’s Theorem independent of Schur-Zassenhaus’). In Section 4.3 some elementary results on \(p\)-groups are proved.

In Section 5.1 some basic results of representation theory are proved (Maschke’s Theorem, Schur’s Lemma, Schur’s relations for matrix elements of irreducible representations). In Section 5.2 one proves the first and second orthogonality relations and some of their consequences, and also Burnside’s \(p^aq^b\)-Theorem. In Section 5.3 irreducible characters of direct products are described. The Burnside-Brauer Theorem on powers of a faithful character is presented.

In Chapter 6 some standard results concentrating around Frobenius reciprocity and Clifford’s Theorem, are proved. Itô’s Theorem (\(\chi(1)\) divides \(| G:A|\) for an abelian normal subgroup \(A\) and \(\chi\in\text{Irr}(G)\)) is presented. One proves that nilpotent groups are monomial. In Section 6.4 some Mackey theorems are proved. A few character tables of small groups are constructed. In Section 6.5 Brauer’s induction theorems are proved (the proofs presented are due to Goldschmidt-Isaacs).

In Chapter 7, some algorithms for calculating the character table of a finite group \(G\) are discussed. The first algorithm, that is due to Burnside, is based on the fact that the structure constants of the class algebra of \(G\) determine the character table of \(G\). J. Dixon produced a revised form of Burnside’s algorithm (Section 7.2). G. J. A. Schneider discovered a variation on the algorithm that in many cases speeds it up considerably (Section 7.3).

Chapter 8 is an introduction to the Young theory of representations of \(S_n\).

Chapter 9 is devoted to Frobenius groups. All material is standard. An example of a Frobenius group with nonabelian kernel that is due to N. Itô, is presented (the first such example of order \(7^3\) was given by Otto Ju. Shmidt).

In Chapter 10 some advanced topics are treated. Brauer’s Realization Theorem is proved in Section 10.1. Section 10.2 is an introduction to the Schur index. The Frobenius-Schur indicator is a subject of Section 10.3; some details are given for Frobenius groups.

The book is well written. It contains some fresh material, esp. in Part 1. Some books with related theme that designate to the same audience, are T. Tambour, Introduction to finite groups and their representations [Lund (1991)] and J. L. Alperin and R. B. Bell, Groups and representations [Springer (1995; Zbl 0839.20001)].

Reviewer: Yakov Berkovich (Afula)

### MSC:

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

20C15 | Ordinary representations and characters |

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

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

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