Sur les groupes classiques. (On the classical groups). Nouveau tirage.
(Sur les groupes classiques.)

*(French)*Zbl 0926.20030
Paris: Hermann. 84 p. (1998).

Apart from two pages of addenda (which relate to results obtained in the mid-50’s and include references) this is a reprinting of the first edition (1948; Zbl 0037.01304).

The object of the book under review is the study of the structures of the classical groups. It extends previous results of L. E. Dickson from the case of finite fields to arbitrary fields. The main tool is provided by the theory of Hermitian and quadratic forms due to E. Witt. The first chapter discusses the case of symplectic groups: alternating forms, totally isotropic subspaces, involutive symplectic transformations, symplectic transvections and the structure of the group. The second and third chapters are devoted to the study of orthogonal groups in odd characteristic and in even characteristic assuming that the defect is zero. (In even characteristic the bilinear form attached to the quadratic form is alternating, if its rank is equal to the dimension of the vector space, the author says that we are in defect zero case.) He studies the isotropic subspaces, the orthogonal transformations, the commutator group of the orthogonal group and gives the structure of the latter one. The next chapter gives the structure of the non-zero defect case. The last two chapters consider the cases of unitary groups on commutative and non-commutative fields respectively.

The object of the book under review is the study of the structures of the classical groups. It extends previous results of L. E. Dickson from the case of finite fields to arbitrary fields. The main tool is provided by the theory of Hermitian and quadratic forms due to E. Witt. The first chapter discusses the case of symplectic groups: alternating forms, totally isotropic subspaces, involutive symplectic transformations, symplectic transvections and the structure of the group. The second and third chapters are devoted to the study of orthogonal groups in odd characteristic and in even characteristic assuming that the defect is zero. (In even characteristic the bilinear form attached to the quadratic form is alternating, if its rank is equal to the dimension of the vector space, the author says that we are in defect zero case.) He studies the isotropic subspaces, the orthogonal transformations, the commutator group of the orthogonal group and gives the structure of the latter one. The next chapter gives the structure of the non-zero defect case. The last two chapters consider the cases of unitary groups on commutative and non-commutative fields respectively.

Reviewer: A.-M.Aubert (Paris)

##### MSC:

20G15 | Linear algebraic groups over arbitrary fields |

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

20Gxx | Linear algebraic groups and related topics |