##
**The classical groups and K-theory. Foreword by J. Dieudonné.**
*(English)*
Zbl 0683.20033

Grundlehren der Mathematischen Wissenschaften, 291. Berlin etc.: Springer-Verlag. xv, 576 p. DM 198.00 (1989).

Classical groups over fields are the general linear groups, orthogonal groups, symplectic groups, unitary groups, and closely related groups. They can be found in many textbooks on linear algebra. Their importance in finite group theory stems from the fact that “most” of the finite simple groups are classical groups over finite fields. In early examples, the field was a finite prime field (Jordan, 1870). Independently, classical groups over real and complex numbers appeared in geometry and mathematical physics as symmetry groups. Note that all simple Lie groups, with finitely many exceptions, are classical groups over the real or complex numbers, or Hamilton quaternions. Their theory was initiated by Lie, Killing, E. Cartan, and their importance in geometry was outlined by F. Klein in his Erlanger program (1872). Dickson (1901) studied classical groups over an arbitrary field, and he showed the essential simplicity of those groups. H. Weyl coined the phrase “classical groups” in his famous book (1946).

A. Weil (1960) showed that “most” of the simple algebraic groups over a field can be realized as classical groups over finite-dimensional algebras over this field. His work was completed (in characteristic 2 case) by Tits. Dieudonné (1943) initiated the study of classical groups over division rings. Later, classical groups over local rings and more general rings appeared in various publications. Bass (1964) studied the general linear groups over an arbitrary associative ring, and he initiated “algebraic K-theory”. Later he studied other classical groups and developed an algebraic framework of L-theory. Since then the theory has been expended to include more general groups.

The book under review gives an excellent account of the fundamental algebraic properties of the classical groups over rings. It discusses basic definitions, the structure, and isomorphisms of these groups. It introduces functors \(K_ 1\), \(K_ 2\), and their unitary analogues. It contains a wealth of recent results, some of them by the authors and their students. The presentation is excellent. The 21-page bibliography includes most of the relevant authors.

A. Weil (1960) showed that “most” of the simple algebraic groups over a field can be realized as classical groups over finite-dimensional algebras over this field. His work was completed (in characteristic 2 case) by Tits. Dieudonné (1943) initiated the study of classical groups over division rings. Later, classical groups over local rings and more general rings appeared in various publications. Bass (1964) studied the general linear groups over an arbitrary associative ring, and he initiated “algebraic K-theory”. Later he studied other classical groups and developed an algebraic framework of L-theory. Since then the theory has been expended to include more general groups.

The book under review gives an excellent account of the fundamental algebraic properties of the classical groups over rings. It discusses basic definitions, the structure, and isomorphisms of these groups. It introduces functors \(K_ 1\), \(K_ 2\), and their unitary analogues. It contains a wealth of recent results, some of them by the authors and their students. The presentation is excellent. The 21-page bibliography includes most of the relevant authors.

Reviewer: L.Vaserstein and E.Wheland

### MSC:

20G35 | Linear algebraic groups over adèles and other rings and schemes |

18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |

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

20F28 | Automorphism groups of groups |

20E32 | Simple groups |