On the theory of soluble Lie algebras

Frattini subalgebras of a class of solvable Lie algebras

“…That Cartan subalgebras are complements to 1/ is shown in (1) implies (2). This completes the proof of Proposition 1.…”

“…Let C be a Cartan subalgebra of M. By Lemma 4 of [1],.C is a Cartan subalgebra of L. If L is a solvable Lie algebra it has been shown in [2] that φ(L) is an ideal of L. We look for a condition on the subalgebras of L/ψ(L) which are necessary and sufficient that Leϊ. In order to do this the following concept is introduced.…”

Minimal Nonnilpotent Solvable Lie Algebras

“…P. Hall and G. Higman have shown in [5] that a nonnilpotent finite group G all of whose proper subgroups are nilpotent can be considered as the product of subgroups P and Q where P is cyclic of prime power order, pa, Q is an invariant g-subgroup of G, q^p, 4>(P) ^Z(G) and Q is either elementary abelian or <j>(Q) = Z(Q) = [Q, Q] where in either case P acts irreducibly on Q/<j>(Q). We shall find a result on solvable Lie algebras which is roughly analogous to this result.Let L be a finite-dimensional solvable Lie algebra and let M be a self-normalizing maximal subalgebra of L. The maximal ideal of L contained in M will be called the core of M. The intersection of all maximal subalgebras of L will be denoted by <p(L) and is an ideal in L by Lemma 3.4 of [2]. The derived algebra of L will be denoted by L' and the center of L by Z(L).…”

“…Let L be a finite-dimensional solvable Lie algebra and let M be a self-normalizing maximal subalgebra of L. The maximal ideal of L contained in M will be called the core of M. The intersection of all maximal subalgebras of L will be denoted by <p(L) and is an ideal in L by Lemma 3.4 of [2]. The derived algebra of L will be denoted by L' and the center of L by Z(L).…”

Ado-Iwasawa extras

“…The theory of saturated formations, set out in Barnes and Gastineau-Hills [5] and of -hypercentral modules, set out in Barnes [1], provides a means of generalising this. 408 Donald W. Barnes [2] A saturated formation of soluble Lie algebras over F is a class # of finite-dimensional soluble Lie algebras over F such that (1) ifL e # a n d A < L , t h e n L / A e £; If L is a soluble Lie algebra over a field F of characteristic 0, then L' is nilpotent.…”