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.