On powerful and p-central restricted Lie algebras

In this note we analyse the analogy between m-potent and p-central restricted Lie algebras and p-groups. For restricted Lie algebras the notion of m-potency has stronger implications than for p-groups (Theorem A). Every finite-dimensional restricted Lie algebra L is isomorphic to (L) over bar/(L) over bar ([p]) for some finite-dimensional p-central restricted Lie algebra (L) over bar (Proposition B). In particular, for restricted Lie algebras there does not hold an analogue of J.Buckley's theorem. For p odd one can characterise powerful restricted Lie algebras in terms of the cup product map in the same way as for finite p-groups (Theorem C). Moreover, the p-centrality of the finite-dimensional restricted Lie algebra L has a strong implication on the structure of the cohomology ring H-. (L, F) (Theorem D).