In this paper we investigate the relation between the multiplicities of split
strongly abelian p-chief factors of finite-dimensional restricted Lie algebras
and first degree restricted cohomology. As an application we obtain a
characterization of solvable restricted Lie algebras in terms of the
multiplicities of split strongly abelian p-chief factors. Moreover, we derive
some results in the representation theory of restricted Lie algebras related to
the principal block and the projective cover of the trivial irreducible module
of a finite-dimensional restricted Lie algebra. In particular, we obtain a
characterization of finite-dimensional solvable restricted Lie algebras in
terms of the second Loewy layer of the projective cover of the trivial
irreducible module. The analogues of these results are well known in the
modular representation theory of finite groups.Comment: 10 pages. arXiv admin note: substantial text overlap with
arXiv:1206.366
