##
**The finite simple groups.**
*(English)*
Zbl 1203.20012

Graduate Texts in Mathematics 251. London: Springer (ISBN 978-1-84800-987-5/hbk; 978-1-84800-988-2/ebook). xv, 298 p. (2009).

This book is aimed at providing an overview of all the finite simple groups in one volume. The author intends to describe the groups in as much detail as possible within 300 pages, including concrete construction, calculation of the order and proof of simplicity, and discusses their actions on various geometrical, algebraic or combinatorial objects in order to study their properties. The author emphasizes connections between exceptional behavior of generic groups and the existence of sporadic groups.

After the publication of M. Aschbacher and S. D. Smith [The classification of quasithin groups. I, II. Mathematical Surveys and Monographs 111, 112. Providence, AMS (2004; Zbl 1065.20023, Zbl 1065.20024)], it has been widely believed that the proof of the classification of finite simple groups is complete. D. Gorenstein, R. Lyons and R. Solomon [The classification of the finite simple groups. Mathematical Surveys and Monographs, 40(1). Providence, AMS (1994; Zbl 0816.20016), 40(2) (1996; Zbl 0836.20011), 40(3) (1998; Zbl 0890.20012), 40(4) (1999; Zbl 0922.20021), 40(5) (2002; Zbl 1006.20012), 40(6) (2005; Zbl 1069.20011)] intend to present the whole proof of the classification theorem. However, the work is still in progress. The current status of the classification theorem can be found in Section 1.4 of the book.

The non-Abelian finite simple groups are divided into four types, namely,

(i) alternating groups \(A_n\),

(ii) classical groups: linear \(\mathrm{PSL}_n(q)\), \(n\geq 2\), except \(\mathrm{PSL}_2(2)\) and \(\mathrm{PSL}_2(3)\); unitary \(\mathrm{PSU}_n(q)\), \(n\geq 3\), except \(\mathrm{PSU}_3(2)\); symplectic \(\mathrm{PSp}_{2n}(q)\), \(n\geq 2\), except \(\mathrm{PSp}_4(2)\); orthogonal \(\text{P}{\Omega}_{2n+1}(q)\), \(n\geq 3\), \(q\) odd; \(\text{P}\Omega_{2n}^+(q)\), \(n\geq 4\); \(\text{P}\Omega_{2n}^-(q)\), \(n\geq 4\),

(iii) exceptional groups of Lie type: \(G_2(q)\), \(q\geq 3\), \(E_6(q)\), \(E_7(q)\), \(E_8(q)\), \(F_4(q)\), \(^3D_4(q)\), \(^2E_6(q)\), \(^2B_2(2^{2n+1})\), \(n\geq 1\), \(^2G_2(3^{2n+1})\), \(n\geq 1\), \(^2F_4(2^{2n+1})\), \(n\geq 1\), \(^2F_4(2)'\),

(iv) 26 sporadic groups: the five Mathieu groups \(M_{11}\), \(M_{12}\), \(M_{22}\), \(M_{23}\), \(M_{24}\); the seven Leech lattice groups \(Co_1\), \(Co_2\), \(Co_3\), \(McL\), \(HS\), \(Suz\), \(J_2\); the three Fischer groups \(Fi_{22}\), \(Fi_{23}\), \(Fi_{24}'\); the five Monstrous groups \(\mathbb{M}\), \(\mathbb{B}\), \(Th\), \(HN\), \(He\); the six pariahs \(J_1\), \(J_3\), \(J_4\), \(O'N\), \(Ly\), \(Ru\).

Among the simple groups in the list, there are exceptional isomorphisms as follows: \(\mathrm{PSL}_2(4)\cong\mathrm{PSL}_2(5)\cong A_5\), \(\mathrm{PSL}_2(7)\cong\mathrm{PSL}_3(2)\), \(\mathrm{PSL}_2(9)\cong A_6\), \(\mathrm{PSL}_4(2)\cong A_8\), and \(\mathrm{PSU}_4(2)\cong\mathrm{PSp}_4(3)\).

The book under review consists of five chapters. Chapter 1 is Introduction, which includes sections on the classification theorem, remarks on the proof of the classification theorem and how to read this book.

The author begins with the alternating groups in Chapter 2. Various properties of the alternating groups such as simplicity, determination of the outer automorphism group, construction of the non-split central extension and certain general subgroups are discussed. The author points out exceptional behavior of the symmetric group \(S_6\) of degree \(6\) in the outer automorphism groups and that of the alternating groups \(A_6\) and \(A_7\) of degree \(6\) and \(7\) in the non-split central extensions. A proof of the O’Nan-Scott theorem concerning maximal subgroups of the symmetric groups is supplied. Moreover, the notion of reflection groups is introduced as a generalization of the symmetric groups, which are important both for the theory of groups of Lie type and for many sporadic groups. The classification of indecomposable finite real reflection groups by Coxeter is also explained.

The subsequent two chapters are devoted to the simple groups of Lie type. A remarkable feature is that the use of Lie algebras is kept minimal in describing those groups. Thus, it is different from R. W. Carter [Simple groups of Lie type. Pure and Applied Mathematics. Vol. XXVIII. London etc.: John Wiley & Sons (1972; Zbl 0248.20015)].

In Chapter 3 the author deals with classical groups, each of which is of the form \(G'/Z(G')\) with \(G\) a matrix group. Simplicity, the automorphism groups, certain subgroups and non-split central extensions of classical groups are studied. First, the author considers the general linear groups over finite fields. Their subgroups and flag varieties of underlying vector spaces are discussed. The notion of projective space is introduced. Using it the author proves the exceptional isomorphisms \(\mathrm{PSL}_2(4)\cong\mathrm{PSL}_2(5)\cong A_5\) and \(\mathrm{PSL}_2(9)\cong A_6\). Next, the author proceeds to bilinear, sesquilinear and quadratic forms which are used for the description of symplectic, unitary and orthogonal groups. A proof of a version of the Aschbacher-Dynkin theorem on maximal subgroups of classical groups is presented. The explicit statement for each family of classical groups is taken from P. Kleidman and M. Liebeck [The subgroup structure of the finite classical groups. London Mathematical Society Lecture Note Series 129. Cambridge etc.: Cambridge University Press (1990; Zbl 0697.20004)]. Furthermore, exceptional isomorphisms among low-dimensional classical groups are explained.

In Chapter 4 the ten families of the exceptional groups of Lie type are discussed. There are three ways to describe those groups, namely, the approaches via Lie algebras, via algebraic groups and via other algebras such as the octonion algebra and the exceptional Jordan algebra. The author takes the third one, an unconventional one for the sake of concrete calculations, and thus treats these ten families separately. First, the author gives an elementary description of the Suzuki group \(^2B_2(2^{2n+1})\) as a group of \(4\times 4\) matrices over the field \(\mathrm{GF}(2^{2n+1})\) of \(2^{2n+1}\) elements. Next, the author introduces the octonion algebras and studies the group \(G_2(q)\). In particular, the exceptional isomorphism \(G_2(2)\cong\mathrm{PSU}_3(3):2\) is proved by using an octonion algebra on the \(E_8\)-lattice. An octonion algebra in characteristic \(3\) is also used to construct \(^2G_2(3^{2n+1})\) and show the isomorphism \(^2G_2(3)\cong\mathrm{PSL}_2(8):3\). Moreover, the group \(^3D_4(q)\) is constructed by using the twisted octonion algebra. As to the group \(F_4(q)\), the author constructs it as the automorphism group of an exceptional Jordan algebra. Various properties of \(F_4(q)\) are presented. Then the author proceeds to the group \(^2F_4(2^{2n+1})\). The group \(E_6(q)\) is constructed as the automorphism group of a cubic form, a similar construction as Dickson’s original one. The remaining groups \(^2E_6(q)\), \(E_7(q)\) and \(E_8(q)\) are briefly mentioned.

Chapter 5, the final chapter, is devoted to the 26 sporadic simple groups. First, the author treats the five Mathieu groups in detail, together with the binary and ternary Golay codes and Steiner systems. Then the author proceeds to the Leech lattice and the seven members of Leech lattice groups. One section is used to explain the Suzuki chain. Next, the author describes the three Fischer groups as automorphism groups of the graphs whose vertices consist of transpositions. Parker’s loop is also discussed. Up to here the proofs for various properties of those sporadic groups are basically supplied. As to the remaining sporadic groups, namely, the five members of Monstrous groups and the six members of pariahs, the author restricts himself to state some important properties of the groups and indicate the outline of their proofs.

This book is a unique introductory overview of all the finite simple groups, and thus it is suitable not only for specialists who are interested in finite simple groups but also for advanced undergraduate and graduate students in algebra. The section entitled ‘Further reading’ at the end of each chapter is a nice guide to further study of the subjects.

After the publication of M. Aschbacher and S. D. Smith [The classification of quasithin groups. I, II. Mathematical Surveys and Monographs 111, 112. Providence, AMS (2004; Zbl 1065.20023, Zbl 1065.20024)], it has been widely believed that the proof of the classification of finite simple groups is complete. D. Gorenstein, R. Lyons and R. Solomon [The classification of the finite simple groups. Mathematical Surveys and Monographs, 40(1). Providence, AMS (1994; Zbl 0816.20016), 40(2) (1996; Zbl 0836.20011), 40(3) (1998; Zbl 0890.20012), 40(4) (1999; Zbl 0922.20021), 40(5) (2002; Zbl 1006.20012), 40(6) (2005; Zbl 1069.20011)] intend to present the whole proof of the classification theorem. However, the work is still in progress. The current status of the classification theorem can be found in Section 1.4 of the book.

The non-Abelian finite simple groups are divided into four types, namely,

(i) alternating groups \(A_n\),

(ii) classical groups: linear \(\mathrm{PSL}_n(q)\), \(n\geq 2\), except \(\mathrm{PSL}_2(2)\) and \(\mathrm{PSL}_2(3)\); unitary \(\mathrm{PSU}_n(q)\), \(n\geq 3\), except \(\mathrm{PSU}_3(2)\); symplectic \(\mathrm{PSp}_{2n}(q)\), \(n\geq 2\), except \(\mathrm{PSp}_4(2)\); orthogonal \(\text{P}{\Omega}_{2n+1}(q)\), \(n\geq 3\), \(q\) odd; \(\text{P}\Omega_{2n}^+(q)\), \(n\geq 4\); \(\text{P}\Omega_{2n}^-(q)\), \(n\geq 4\),

(iii) exceptional groups of Lie type: \(G_2(q)\), \(q\geq 3\), \(E_6(q)\), \(E_7(q)\), \(E_8(q)\), \(F_4(q)\), \(^3D_4(q)\), \(^2E_6(q)\), \(^2B_2(2^{2n+1})\), \(n\geq 1\), \(^2G_2(3^{2n+1})\), \(n\geq 1\), \(^2F_4(2^{2n+1})\), \(n\geq 1\), \(^2F_4(2)'\),

(iv) 26 sporadic groups: the five Mathieu groups \(M_{11}\), \(M_{12}\), \(M_{22}\), \(M_{23}\), \(M_{24}\); the seven Leech lattice groups \(Co_1\), \(Co_2\), \(Co_3\), \(McL\), \(HS\), \(Suz\), \(J_2\); the three Fischer groups \(Fi_{22}\), \(Fi_{23}\), \(Fi_{24}'\); the five Monstrous groups \(\mathbb{M}\), \(\mathbb{B}\), \(Th\), \(HN\), \(He\); the six pariahs \(J_1\), \(J_3\), \(J_4\), \(O'N\), \(Ly\), \(Ru\).

Among the simple groups in the list, there are exceptional isomorphisms as follows: \(\mathrm{PSL}_2(4)\cong\mathrm{PSL}_2(5)\cong A_5\), \(\mathrm{PSL}_2(7)\cong\mathrm{PSL}_3(2)\), \(\mathrm{PSL}_2(9)\cong A_6\), \(\mathrm{PSL}_4(2)\cong A_8\), and \(\mathrm{PSU}_4(2)\cong\mathrm{PSp}_4(3)\).

The book under review consists of five chapters. Chapter 1 is Introduction, which includes sections on the classification theorem, remarks on the proof of the classification theorem and how to read this book.

The author begins with the alternating groups in Chapter 2. Various properties of the alternating groups such as simplicity, determination of the outer automorphism group, construction of the non-split central extension and certain general subgroups are discussed. The author points out exceptional behavior of the symmetric group \(S_6\) of degree \(6\) in the outer automorphism groups and that of the alternating groups \(A_6\) and \(A_7\) of degree \(6\) and \(7\) in the non-split central extensions. A proof of the O’Nan-Scott theorem concerning maximal subgroups of the symmetric groups is supplied. Moreover, the notion of reflection groups is introduced as a generalization of the symmetric groups, which are important both for the theory of groups of Lie type and for many sporadic groups. The classification of indecomposable finite real reflection groups by Coxeter is also explained.

The subsequent two chapters are devoted to the simple groups of Lie type. A remarkable feature is that the use of Lie algebras is kept minimal in describing those groups. Thus, it is different from R. W. Carter [Simple groups of Lie type. Pure and Applied Mathematics. Vol. XXVIII. London etc.: John Wiley & Sons (1972; Zbl 0248.20015)].

In Chapter 3 the author deals with classical groups, each of which is of the form \(G'/Z(G')\) with \(G\) a matrix group. Simplicity, the automorphism groups, certain subgroups and non-split central extensions of classical groups are studied. First, the author considers the general linear groups over finite fields. Their subgroups and flag varieties of underlying vector spaces are discussed. The notion of projective space is introduced. Using it the author proves the exceptional isomorphisms \(\mathrm{PSL}_2(4)\cong\mathrm{PSL}_2(5)\cong A_5\) and \(\mathrm{PSL}_2(9)\cong A_6\). Next, the author proceeds to bilinear, sesquilinear and quadratic forms which are used for the description of symplectic, unitary and orthogonal groups. A proof of a version of the Aschbacher-Dynkin theorem on maximal subgroups of classical groups is presented. The explicit statement for each family of classical groups is taken from P. Kleidman and M. Liebeck [The subgroup structure of the finite classical groups. London Mathematical Society Lecture Note Series 129. Cambridge etc.: Cambridge University Press (1990; Zbl 0697.20004)]. Furthermore, exceptional isomorphisms among low-dimensional classical groups are explained.

In Chapter 4 the ten families of the exceptional groups of Lie type are discussed. There are three ways to describe those groups, namely, the approaches via Lie algebras, via algebraic groups and via other algebras such as the octonion algebra and the exceptional Jordan algebra. The author takes the third one, an unconventional one for the sake of concrete calculations, and thus treats these ten families separately. First, the author gives an elementary description of the Suzuki group \(^2B_2(2^{2n+1})\) as a group of \(4\times 4\) matrices over the field \(\mathrm{GF}(2^{2n+1})\) of \(2^{2n+1}\) elements. Next, the author introduces the octonion algebras and studies the group \(G_2(q)\). In particular, the exceptional isomorphism \(G_2(2)\cong\mathrm{PSU}_3(3):2\) is proved by using an octonion algebra on the \(E_8\)-lattice. An octonion algebra in characteristic \(3\) is also used to construct \(^2G_2(3^{2n+1})\) and show the isomorphism \(^2G_2(3)\cong\mathrm{PSL}_2(8):3\). Moreover, the group \(^3D_4(q)\) is constructed by using the twisted octonion algebra. As to the group \(F_4(q)\), the author constructs it as the automorphism group of an exceptional Jordan algebra. Various properties of \(F_4(q)\) are presented. Then the author proceeds to the group \(^2F_4(2^{2n+1})\). The group \(E_6(q)\) is constructed as the automorphism group of a cubic form, a similar construction as Dickson’s original one. The remaining groups \(^2E_6(q)\), \(E_7(q)\) and \(E_8(q)\) are briefly mentioned.

Chapter 5, the final chapter, is devoted to the 26 sporadic simple groups. First, the author treats the five Mathieu groups in detail, together with the binary and ternary Golay codes and Steiner systems. Then the author proceeds to the Leech lattice and the seven members of Leech lattice groups. One section is used to explain the Suzuki chain. Next, the author describes the three Fischer groups as automorphism groups of the graphs whose vertices consist of transpositions. Parker’s loop is also discussed. Up to here the proofs for various properties of those sporadic groups are basically supplied. As to the remaining sporadic groups, namely, the five members of Monstrous groups and the six members of pariahs, the author restricts himself to state some important properties of the groups and indicate the outline of their proofs.

This book is a unique introductory overview of all the finite simple groups, and thus it is suitable not only for specialists who are interested in finite simple groups but also for advanced undergraduate and graduate students in algebra. The section entitled ‘Further reading’ at the end of each chapter is a nice guide to further study of the subjects.

Reviewer: Hiromichi Yamada (Tokyo)

### MSC:

20D05 | Finite simple groups and their classification |

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

20D08 | Simple groups: sporadic groups |

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