Three-Dimensional Affine Groups

As shown in the previous sections, three-dimensional groups which are unimaximal or uni-minimal play a significant rôle in our theory.We now shall give a complete classification of three-dimensional connected unipotent algebraic groups defined over a field k of characteristic p >2. Some of our results even hold in the case p= 2. A main tool is the theory of extensions, which is particularly efficient for unipotent groups defined over a perfect field, as we have seen in Remarks 2.1.4 and 2.1.5. As already mentioned in Section 4.2, we obtain at the same time a classification of the groups of k-rational points of three-dimensional connected unipotent algebraic groups, if the field k is infinite and perfect.

By Corollary 4.2.10, if p >2 and the three-dimensional unipotent group Gis a chain, then G' is one-dimensional, and we can refer to Theorem 4.3.1. Therefore in the present section we consider groups which are not chains.