Some Criteria for Nilpotency in Groups and Lie Algebras

We shall say that an automorphism a is nilpotent or acts nilpotently on a group G if in the holomorph H= [G](a) of G with a, a is a bounded left Engel element, that is, [H, ¿a] = l for some natural number ¿. Here [H, ka] means [H, (k — l)a] with [H, Oa] denoting H. Let G' denote the commutator subgroup [G, G], and let $(G) denote the Frattini subgroup of G. If a is an automorphism of a nilpotent group G such that the automorphism ä induced by a on G/G' is nolpotent (or with certain restrictions on the exponent of G on G/Í»(G)), then by a well-known theorem of Philip Hall (cf. [6, p. 202]), a is nilpotent. Here we shall show that the same conclusion follows if we know that the restriction of a to a suitable subgroup of a nilpotent group is nilpotent. We prove the following two theorems announced in [7].