Schur proved that if the center of a group G has finite index, then the derived subgroup G′ is also finite. Moneyhun proved that if L is a Lie algebra such that dim(L/Z(L)) = n, then dim(L^2) ≤1/2n(n-1) In this paper, we extend the converse of Moneyhun’s theorem. Also, we prove a wellknown result of nilpotent Lie algebras by using a different technique

In this paper, we shall apply symmetry analysis to second order functional di, erential equations with constant coe, cients. The determining equations of the admitted Lie group are constructed in a manner di, erent from that of the existing literature for delay di, erential equations. We de, ne the standard Lie bracket and make a complete classi, cation of the second order linear functional di, erential equations with constant coe, cients, to solvable Lie algebras. We also classify some second order nonlinear functional di, erential equations with constant coe, cients, to solvable Lie algebras.

Let $L$ be a finite dimensional Lie algebra over an arbitrary field $F$. In this paper, we prove that the class of finite nilpotent(solvable) Lie algebras is an example of formation. Furthermore, we conclude that every finite Lie algebra has a nilpotent(solvable) residual. Finally we prove some results on Frattini and Fitting subalgebras of the nilpotent Lie algebra $L$.

This paper is devoted to deformation theory of graded Lie algebras over Z or Zl with finite dimensional graded pieces. Such deformation problems naturally appear in number theory. In the first part of the paper, we use Schlessinger criteria for functors on Artinian local rings in order to obtain universal deformation rings for deformations of graded Lie algebras and their graded representations. In the second part, we use a version of Schlessinger criteria for functors on the Artinian category of nilpotent Lie algebras which is formulated by Pridham, and explore arithmetic deformations using this technique.