Search results for "Additive group"
showing 7 items of 7 documents
Proper triangular Ga-actions on A^4 are translations
2013
We describe the structure of geometric quotients for proper locally triangulable additve group actions on locally trivial A^3-bundles over a noetherian normal base scheme X defined over a field of characteristic 0. In the case where dim X=1, we show in particular that every such action is a translation with geometric quotient isomorphic to the total space of a vector bundle of rank 2 over X. As a consequence, every proper triangulable Ga-action on the affine four space A^4 over a field of characteristic 0 is a translation with geometric quotient isomorphic to A^3.
On the subset sum problem for finite fields
2021
Abstract Let G be the additive group of a finite field. J. Li and D. Wan determined the exact number of solutions of the subset sum problem over G, by giving an explicit formula for the number of subsets of G of prescribed size whose elements sum up to a given element of G. They also determined a closed-form expression for the case where the subsets are required to contain only nonzero elements. In this paper we give an alternative proof of the two formulas. Our argument is purely combinatorial, as in the original proof by Li and Wan, but follows a different and somehow more “natural” approach. We also indicate some new connections with coding theory and combinatorial designs.
Equivariant Triviality of Quasi-Monomial Triangular $$\mathbb{G}_{a}$$-Actions on $$\mathbb{A}^{4}$$
2014
We give a direct and self-contained proof of the fact that additive group actions on affine four-space generated by certain types of triangular derivations are translations whenever they are proper. The argument, which is based on explicit techniques, provides an illustration of the difficulties encountered and an introduction to the more abstract methods which were used recently by the authors to solve the general triangular case.
Rationally integrable vector fields and rational additive group actions
2016
International audience; We characterize rational actions of the additive group on algebraic varieties defined over a field of characteristic zero in terms of a suitable integrability property of their associated velocity vector fields. This extends the classical correspondence between regular actions of the additive group on affine algebraic varieties and the so-called locally nilpotent derivations of their coordinate rings. Our results lead in particular to a complete characterization of regular additive group actions on semi-affine varieties in terms of their associated vector fields. Among other applications, we review properties of the rational counterpart of the Makar-Limanov invariant…
Local nearrings with dihedral multiplicative group
2004
AbstractA not necessarily zero-symmetric nearring R with a unit element is called local if the set of all non-invertible elements of R forms a subgroup of the additive group of R. It is proved that every local nearring whose multiplicative group is dihedral is finite and its additive group is either a 3-group of order at most 9 or a 2-group of order at most 32.
On the adjoint group of some radical rings
1997
A ring R is called radical if it coincides with its Jacobson radical, which means that Rforms a group under the operation a ° b = a + b + ab for all a and b in R. This group is called the adjoint group R° of R. The relation between the adjoint group R° and the additive group R+ of a radical rin R is an interesting topic to study. It has been shown in [1] that the finiteness conditions “minimax”, “finite Prufer rank”, “finite abelian subgroup rank” and “finite torsionfree rank” carry over from the adjoint group to the additive group of a radical ring. The converse is true for the minimax condition, while it fails for all the other above finiteness conditions by an example due to Sysak [6] (s…
Triply Factorised Groups and the Structure of Skew Left Braces
2021
The algebraic structure of skew left brace has proved to be useful as a source of set-theoretic solutions of the Yang–Baxter equation. We study in this paper the connections between left and right $$\pi $$ -nilpotency and the structure of finite skew left braces. We also study factorisations of skew left braces and their impact on the skew left brace structure. As a consequence of our study, we define a Fitting-like ideal of a left brace. Our approach depends strongly on a description of a skew left brace in terms of a triply factorised group obtained from the action of the multiplicative group of the skew left brace on its additive group.