The Generation of Sp([F,) by Transvections

In 1955, in [2], Dieudonne, observing that each symplectic group Sp(V) is generated by its transvections, considered the problem of finding the minimal number of factors in the expression of elements 0 of Sp(V) as products of transvections in Sp(V).l The answer is in general residue cr, but with some classes of exceptions. Complications arise when the underlying field is 5, , and Dieudonne’s list of exceptions [2, p. 1641 for this case is incomplete. The difficulty occurs in 4.4” page 163 and seems to lie in the fact that, following the notation there, the plane T chosen orthogonal to c and d need not (in fact, cannot) be orthogonal to the revised “c” and ‘Id,” as is tacitly assumed. The proof in 4.4” fails and there is a class of exceptions of residue 5 in dimension 6 (see 3.3 below). This paper is devoted to completing the list of exceptions for f fs . Also mentioned are the minor corrections thus necessitated in the theory for the orthogonal group in characteristic 2, treated in [2, p. 1771. Section 1 is introductory, establishing notation and the basic facts needed. Section 2 recasts results of [2]. Section 3 presents the “new” exceptions and Section 4 shows there are no more. Section 5 collects results to state the main theorem. Section 6 considers the orthogonal case.