Among the new results I have obtained, the most striking is probably the fact that, for the orthogonal and unitary groups, there is a sharp distinction to be drawn between the case in which the quadratic (resp. hermltian) form defining the group may be 0 for vectors ф 0, and the case in which it may not. (When the underlying field is the real (resp. complex) field, this means of course that the form is "positive-definite" or "negative-definite.') In the first case, general statements may be made regarding the structure of the group, irrespective of the nature of the underlying field, whereas in the other case examples show that the group may belong to very different types according to the type of field; this curious fact had remained unsuspected up to now, because in the classical case (real field) it turns out that the orthogonal group has similar structure in both cases, and when the underlying field is finite the second case is impossible.