ar X iv : 0 80 9 . 44 43 v 1 [ m at h . G R ] 2 5 Se p 20 08 Algebraic ( 2 , 2 ) - transformation groups ∗

In this paper we determine all algebraic transformation groups G, defined over an algebraically closed field k, which operate transitively, but not primitively, on a variety Ω, provided the following conditions are fulfilled. We ask that the (non-effective) action of G on the variety of blocks is sharply 2-transitive, as well as the action on a block ∆ of the normalizer G∆. Also we require sharp transitivity on pairs (X, Y ) of independent points of Ω, i.e. points contained in different blocks. Although classifications of imprimitive permutation groups appeared already at beginning of the last century (see [10]) and imprimitive actions play an important role in geometry, the corresponding literature is actually less well-developed than the one concerning primitive groups. For finite groups some classification has been done (see for instance [1], [5] and [11]). In [1] by using wreath products, the best-known construction principle to get imprimitive groups, a classification of finite imprimitive groups, acting highly transitively on blocks and satisfying conditions very common in geometry, is achieved. The present paper arises with the aim to obtain classifications for infinite imprimitive groups belonging to well-studied categories. We start with an imprimitive algebraic group G, over an algebraically closed field k, operating on an algebraic variety Ω of positive dimension in such a way that the induced actions on the set Ω of blocks and on a block ∆ are both sharply 2-transitive. Moreover we ask the group to act sharply transitively on pairs of points lying in different blocks. The latter condition, frequently occurring in geometry (see for instance [2]), avoids a too general context. For the classification we do not need the group actions be bi-regular morphisms but we just ask that the orbit maps be separable morphisms. It turns out that G is the semidirect product of a 3-dimensional unipotent connected group Gu by a 1-dimensional connected torus T , both acting on the points of an affine plane over k with a full set of parallel lines as the blocks. There are two subgroups which play a fundamental role for the classification: the kernel G[ Ω ] of the representation on Ω (the so-called inertia subgroup) and its stabilizer G[ Ω ]O of a fixed point O, which turns out to be even the point-wise stabilizer of the block containing O. There exists a G-invariant transversal L of G with respect to G[ Ω ]O which is essential for the classification. L is a subgroup precisely if Gu/z(Gu) This paper contains the more significant part of the article with the same title that will appear in the Volume 12 of Journal of Group Theory