Algebraic (2, 2)-transformation groups

Abstract

Abstract. 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 W, subject to the following conditions. We require that the (non-e¤ective) action of G on the variety of blocks is sharply 2-transitive, as well as the action on a block D of the normalizer G D . Also we require sharp transitivity on pairs ðX ; Y Þ of independent points of W, i.e. points contained in di¤erent blocks.Although classifications of imprimitive permutation groups first appeared at the beginning of the last century (see [10]) and imprimitive actions play an important role in geometry, the corresponding literature is less well developed than that concerning primitive groups. For finite groups there are some classification results (see for instance [1], [5] and [11]). In [1], by using wreath products, the best-known construction principle for obtaining imprimitive groups, a classification is achieved of finite imprimitive groups acting highly transitively on blocks and satisfying conditions very common in geometry.The aim of the present paper is 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 W of positive dimension in such a way that the induced actions on the set W of blocks and on a block D are both sharply 2-transitive. Moreover we require that the group acts sharply transitively on pairs of points lying in di¤erent blocks. The latter condition, frequently occurring in geometry (see for instance [2]), provides a manageable class of groups. For the classification we do not require that the group actions are bi-regular morphisms but merely that the orbit maps be separable morphisms. It turns out that G is the semidirect product of a 3-dimensional unipotent connected group G u by a 1-dimensional connected torus T, both acting on the points of an a‰ne 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 ½W of the representation on W (the so-called inertia subgroup) and its stabilizer G ½W O of a fixed point O, which turns out to be even the pointwise stabilizer of Unauthenticated Download Date | 5/12/18 3:53 AM

Locally compact (2, 2)‐transformation groups

“…In the case i), as well as in the case ii), we have z = e 4 , e 6 , hence u = e 3 , e 5 , because elements not in e 3 are automorphisms of the Lie algebra given in the claim. Moreover, since there are two linearly independent vectors in n which are fixed by K, this vectors must be e 3 and e 4 , hence the above one is the only possible action.…”

“…Although for algebraic groups over an algebraically closed field any sharply 2-transitive group is isomorphic to the group of affine mappings of the line, in the case of positive characteristic there are many imprimitive algebraic groups acting sharply 2-transitively on the system of blocks as well as on any block, as it is shown in [3]. This is essentially due to the existence of the Frobenius automorphism.…”

“…Let U be the subgroup of G/H which acts as homotheties on h/n. Since e 3 , e 5 , e 6 and e 4 , e 5 , e 6 are ideals of the Lie algebra g of G, the elements of U act on n/z by e 3 −it } and these pairs of eigenvalues are distinct, the group G is the semidirect product of H with the group C acting on H as the group of matrices…”

Our Friend and Mathematician Karl Strambach

“…As the manuscript was growing, the young collaborators began to feel the necessity of proving their work with a publication, thus in 2004 (sic!) they came out with the first one [1], and, after five years, with a second one [2]. In the first one, they showed that the case where m > 2 has only few cases.…”

“…Thus, the investigation can be confined to the case of what we called a (2, 2)-group, which can be illustrated as follows: let Ω be a set operated on imprimitively by a group G, in such a way that the stabilizer G Δ of a block Δ acts sharply 2-transitive on Δ, that G/N is sharply 2-transitive on the system of imprimitivity Ω, that is, the set of blocks, where N is the inertia group, that is, the normal subgroup that leaves each block invariant, and, finally, that G is sharply transitive on the set Λ := {(P 1 , P 2 ) ∈ Ω 2 : Δ P1 = Δ P2 }. Algebraic (2, 2)-transformation groups had been characterized in 2009 [2]. In this paper, they exhibit an interesting class of examples where the group G has a non-Abelian, regular, subgroup T of translations: for an algebraically closed field k of characteristic p > 0, this group is defined on the set (k + ) 3 k * by the 1. there is one family of four-dimensional groups, parametrized by a nonzero real number s, which give deformations of the group of affine mappings over the dual numbers over R, which is obtained for s = 1; 2. there is one family of eight-dimensional groups, parametrized by a real number s and an integer number n ≥ 0, and for s = n = 1 one obtains the group of affine mappings of the affine line over the algebra of dual numbers over the complex field.…”