On the theory of soluble Lie algebras

In this paper, we develop for finite-dimensional soluble Lie algebras, a theory of saturated homomorphs and formations and of functors analogous to the theories developed for finite soluble groups by Gaschtitz, Lubeseder, Schunck, Barnes and Kegel. Much of this requires little more than verbal alteration of the group theory form, replacing the words "group" and "normal subgroup" by "algebra" and "ideal" wherever they occur. Even where this translation is completely routine, for the sake of completeness, we give at least brief indications of the proofs. All algebras in the following are finite-dimensional soluble Lie algebras over some fixed ground field F. Except where specifically stated, no assumptions are made about the field F. We write A c L, A <L, A ~ L for A is a subset, subalgebra, ideal of L respectively. If A_~ L, we denote by <A) the subspace spanned by A. If A <L, we denote the normaliser and centraliser of A in L by ~ (A) and cg L (A) respectively. We denote the dimension of L by dim L.