On saturated formations of solvable Lie algebras

Abstract

The concepts of formations, J^~-projectors and J^~normalizers have all been developed for solvable Lie algebras. In this note, for each saturated formation J^~ of solvable Lie algebras, the class ^Γ(κ^) of solvable Lie algebras L in which each of L is an ^'-projector is considered. This is the natural generalization of the Lie algebra analogue to SC groups which were first investigated by R. Carter. It is shown that J7~(J^) is a formation. Then some properties of J^Vnormalizer s of Le/(^) are considered. All Lie algebras considered here are solvable and finite dimensional over a field F. J^~ will always denote a saturated formation of solvable Lie algebras and L will be a solvable Lie algebra. N(L) is the nil-radical of L and Φ(L) is the Frattini subalgebra of L. For definitions and properties of all these concepts see [3], [4], and [9]. For SC groups see [6]. We begin with a general lemma. LEMMA 1. Let N be an ideal of L and D/N be an J^~-normalizer of L/N. Then there exists an ^-normalizer E of L such that E + N= D. Proof. Let L be a minimal counterexample and we may assume