APPLICATIONS OF LIE RINGS WITH FINITE CYCLIC GRADING E. I. KHUKHRO

Ln−1 Let L = i=0 Li be a (Z/nZ)-graded Lie ring (algebra), where the Li are additive subgroups (subspaces) satisfying [Li , Lj ] ⊆ Li+j (mod n) . Theorems of Higman, Kostrikin, and Kreknin assert that if L0 = 0, then L is soluble (for n prime, nilpotent) of n-bounded (i. e. bounded in terms of n) derived length (class). Hence the same follows for a Lie ring M with a regular (i. e. without nontrivial fixed points) automorphism ϕ of order n: after adjoining a primitive nth root of unity ω we obtain M = M0 + M1 + · · · + Mn−1 for Mi = {x ∈ M | ϕ(x) = ω i x}, where [Mi , Mj ] ⊆ Mi+j (mod n) and M0 = 0 (the fact that the sum is not direct in general is inessential). A similar assertion easily follows for (locally) nilpotent groups with a regular automorphism of prime order. But there is an open problem whether an analogue of Kreknin’s theorem holds for such groups with a regular automorphism of arbitrary finite order. Nevertheless, Kreknin’s theorem was successfully applied to finite p-groups with an automorphism of order pk and to pro-p-groups of given coclass in the papers of Jaikin-Zapirain, Khukhro, Medvedev, Shalev, Shalev–Zel’manov. Makarenko and Khukhro proved that if dim L0 = r (or |L0 | = r), then L contains a soluble (for n prime, nilpotent) ideal of n-bounded derived length (nilpotency class) and of (n, r)-bounded codimension. Khukhro applied this result to Lie rings and periodic nilpotent groups with an “almost regular” automorphism of prime order n; Medvedev lifted the periodicity condition for groups. Suppose that there are only d nonzero components among the grading components Li . Shalev and Khukhro proved that is L0 = 0, then L is soluble (for n prime, nilpotent) of d-bounded derived length (class). These results were applied to groups of bounded rank with almost regular automorphisms. In the works of Makarenko, Khukhro, Shumyatsky the condition L0 = 0 was replaced by dim L0 = r (or |L0 | = r): then L contains a soluble (for n prime, nilpotent) ideal of d-bounded derived length (class) and of (d, r)-bounded codimension. There results were applied to generalize Jacobson’s theorem on Lie algebras with a nilpotent algebra of derivations to the case of “almost without nontrivial constants”. Suppose that for some m for k 6= 0 we have |{i | [Lk , Li ] 6= 0}| ≤ m, i. e. each component Lk for k 6= 0 commutes with all but at most m components. Khukhro proved that if L0 = 0, then L is soluble (for n prime, nilpotent) of m-bounded derived length (class), and if dim L0 = r (or |L0 | = r), then L contains a soluble (for n prime, nilpotent) ideal of m-bounded derived length (class) and of (n, r)-bounded codimension. These results were applied to nilpotent groups with Frobenius groups of automorphisms. Sobolev Institute of Mathematics, Novosibirsk E-mail address: [email protected]