A FRATTINI THEORY FOR ALGEBRAS By D. A. TOWERS [Received 11 May 1972] 1. Introduction and outline The theory of the Frattini subgroup of a group is well developed and has proved useful in the study of certain types of problem in group theory. Analogous problems for algebras can be posed, and these are of indepen- dent interest. It therefore seems desirable to investigate the possibility of establishing a parallel theory for algebras. The many close connections which Lie algebras have with groups render them the obvious choice for a first attempt at an analogous theory, and such investigations have been successfully carried out by Barnes ([4]), Barnes and Gastineau- Hills ([6]), Barnes and Newell ([7]), Chao ([8]), Marshall ([17]), Schwarck ([19]), and Stitzinger ([21] and [22]). For associative algebras the Frattini subalgebra has been mentioned by Stitzinger ([20]) and used by Knopfmacher ([16]) to solve some number-theoretic type of problems concerned with finite algebras. However, no exten- sive theory has appeared for any class of algebras apart from Lie algebras. The purpose of the present paper is to develop a Frattini theory for general nonassociative algebras, although we often restrict our attention to Lie or associative algebras in order to obtain deeper structure theorems. Some of the preliminary work requires little more than a translation from the language of group theory to that of the theory of algebras, and proofs will not be given. The main interest naturally centres on the results requiring different proofs, and in the areas where the theory for algebras diverges from that for groups. The definition and several elementary properties of the Frattini subalgebra are collated in §2. We then study in §3 the fundamental question of invariance; that is, whether the Frattini subalgebra is always an ideal. Using the theory of algebraic groups as expounded by Chevalley in [9] and [10], it is shown that this is the case for Lie algebras over fields of characteristic zero (a result pointed out, but not proved, by Barnes in [5]), thereby extending a result of Marshall ([17]). It is seen also, however, that the result does not hold generally, and as a consequence the Frattini ideal is introduced. Proc. London Math. Soc. (3) 27 (1973) 440-462