ON TORSION-FREE ABELIAN GROUPS AND LIE ALGEBRAS RICHARD BLOCK It is known that many of the classes of simple Lie algebras of prime characteristic of nonclassical type have simple infinite-dimensional analogues of characteristic zero (see, for example, [4, p. 518]). We consider here analogues of those algebras which are defined by a modification of the definition of a group algebra. Thus we consider analogues of the Zassenhaus algebras as generalized by Albert and Frank in [l]. The algebras considered are defined as follows. Let G be a nonzero abelian group, F a field, g an additive mapping from G to F, and fan alternate biadditive mapping from GXG to F. We index a basis of an algebra over F by G, denoting by ua the basis element correspond- ing to a in G, and define multiplication by (1) uaue = {f(a, P) + g(a - 0)} ua+li. We designate this algebra by L(G, g,f). We shall determine necessary and sufficient conditions on/and g for the algebra L(G, g, f) to be a simple Lie algebra. We shall then consider the case in which L(G, g,f) is a simple Lie algebra of characteristic zero. This will be seen to imply that G is torsion-free. The derivations and locally algebraic derivations of L(G, g, f) will be determined in this case. Using these, we shall show that any one of these simple Lie algebras L(G, g, f) of characteristic zero determines the group G up to isomorphism and determines the mappings g and / up to a scalar multiple. Our proof of the simplicity of L(G, g, f) and determination of the derivations of L(G, g,f) are also valid when Ehas prime characteristic p and G is an elementary abelian ^-group. However in that case our method for showing that L(G, g, f) essentially determines G, g and / cannot be used—indeed, Ree showed in [4] that all Zassenhaus alge- bras of dimension pn over F are isomorphic. When the torsion-free abelian group G has rank one, the simple algebra L(G, g, f) over F, of characteristic zero, is isomorphic to the algebra of derivations of the group algebra of G over F. Thus the group algebra of a torsion-free abelian group of rank one determines the group. However this is a special case of a result that follows from Higman's determination of the units of group algebras in [2]. Presented to the Society January 29, 1958, under the title of Relations between torsion-free abelian groups and certain Lie algebras; received by the editors March 10, 1958. 613 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use