We classify all (finitely dimensional nilpotent Lie k-algebras h with 2-dimensional commutator ideals h', extending a known result to the case where h' is non-central and k is an arbitrary field. It turns\ud out that, while the structure of h depends on the field k if h' is central, it is independent of k if h' is non-central and is uniquely determined by the dimension of h. In the case where k is algebraically or real closed, we also list all nilpotent Lie k-algebras h with 2-dimensional central commutator ideals h' and dimk h < 12

We investigate derivations of nilpotent complex Lie algebras of type {2n, 1, 1} with the aim to classify solvable complex Lie algebras the commutator ideals of which have codimension 1 and are nilpotent Lie algebras of type {2n, 1, 1}.

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

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We give a representation of any integer as a vector of the Witt ring W(Z_p) and relate it to the Fermat quotient q(n) = (n^(p−1) − 1)/p. Logarithms are introduced in order to establish an isomorphism between the commutative unipotent groups 1+ pW(Z_p) and W(Z_p)