Frattini subalgebras of a class of solvable lie algebras

Abstract

In this paper the Lie algebra analogues to groups with property E of Bechtell are investigated. Let X be the class of solvable Lie algebras with the following property: if H is a subalgebra of L, then (formula presented) where φ(L) denotes the Frattini subalgebra of L; that is, φ(L) is the intersection of all maximal subalgebras of L. Groups with the analogous property are called ingroups by Bechtell. The class X is shown to contain all solvable Lie algebras whose derived algebra is nilpotent. Necessary conditions are found such that an ideal N of L ∈X be the Frattini subalgebra of L. Only solvable Lie algebras of finite dimension are considered here. © 1970 by Pacific Journal of Mathematics.