A Note on Derivations of Lie Algebras

In this note, we will prove that a finite dimensional Lie algebra $L$ of characteristic zero, admitting an abelian algebra of derivations $D\leq Der(L)$ with the property $$L^n\subseteq \sum_{d\in D}d(L)$$ for some $n\geq 1$, is necessarily solvable. As a result, if $L$ has a derivation $d:L\to L$, such that $L^n\subseteq d(L)$, for some $n\geq 1$, then $L$ is solvable.Comment: 4 page

Topo-Groups and a Tychonoff Type Theorem

In this article, we introduce an interesting topology-like concept concerning groups (and with almost the same method it can be defined for other algebraic systems). Given an arbitrary group $G$, we define a {\em topo-system} on $G$ as a set of subgroups satisfying certain conditions like a topology on a set. We will call such a group, a {\em topo-group}. These topo-groups are not rare and as we will see, there are many examples of topo-groups. We investigate fundamental notions concerning topo-groups and we see that many basic concepts of topology can be introduced in the frame of topo-groups. A {\em filter of subgroups} will be defined in such way that we will be able to formulate a Tychonoff type theorem for the direct product of {\em topo-compact} topo-groups.Comment: 9 pages, Revise

Existentially closed structures and some embedding theorems

Using the notion of existentially closed structures, we obtain embedding theorems for groups and Lie algebras. We also prove the existence of some groups and Lie algebras with prescribed properties.Comment: 14 pages, 2 new sections are added, some corrections mad