Frattini Subalgebras of Finitely Generated Soluble Lie Algebras

Abstract

This paper is motivated by a recent one of Stewart and Towers [8] investigating Lie algebras with "good Frattini structure" (definition below).One consequence of our investigations is to prove that any finitely generated metanilpotent Lie algebra has good Frattini structure, thereby answering a question of Stewart and Towers and providing a complete Lie theoretic analogue of the corresponding group theoretic result of Phillip Hall.It will also be shown that in prime characteristic, finitely generated nilpotent-by-finite-dimensional Lie algebras have good Frattini structure.1. Preliminaries.We employ the notation of Amayo and Stewart [3].For a fixed ground field f, 21, 2r» ©» 9Î denote the classes of abelian, finite-dimensional, finitely generated, and nilpotent Lie, algebras respectively.If 3E and ?J are classes of Lie algebras, then 3E2) is the class of all Lie algebras L having an ideal / G jE with L/I E 3J.We write £2 for the class 7¡3L, and, in general, £"+' = £"£.We also refer to £2J as the class of X-by-3J Lie algebras, and X2 is the class of meta-ï algebras.Thus 9Î2 is the class of metanilpotent Lie algebras.The symbol L will denote a Lie algebra of arbitrary dimension defined over the field f.The notation A C L, A < L, A < L, A si L means that A is a subset, subalgebra, ideal, and subideal of L, respectively.By A <-L we mean that A is a maximal subalgebra of L. If A, B C L, then [A, B] is the subspace of L spanned by all [a, b] with a E A and b E B, [A,H+XB] = [[A,aB], B] and [A,0B] -A; [a,0b] = a and [a,n+xb] = [[a,nb], b].The Frattini subalgebra F{L) is the intersection of the maximal subalgebras of L or is L if there are no maximal subalgebras.The Frattini ideal $(L) is the largest ideal of L contained in F{L).In general, F{L) ¥= *(L).A chief factor of L is a pair (H, K) of ideals of L such that H > K and no ideal of L lies properly between H and K.We also refer to the corresponding factor ideal H/K of L/K as the chief factor.If A < B < L, then