Frattini theory for restricted Lie algebras
For restricted Lie algebras the Frattini theory is developed. By \(F_p(L)\) the intersection of all maximal \(p\)-subalgebras of a restricted Lie algebra \(L\) is denoted, and \(\Phi_p(L)\) is the largest \(p\)-ideal of \(L\) contained in \(F_p(L)\). The main results are: 1) For any restricted algebra \(L\) with nilradical \(N\), \(\Phi_p(L)=F_p(L)\cap N\); 2. The restricted algebra \(L\) is \(\Phi_p\)-free if and only if it \(p\)-splits over its abelian \(p\)-socle. 3. If a restricted algebra \(L\) is \(\Phi_p\)-free then the nilradical of \(L\) is the abelian \(p\)-socle.
scientific article; zbMATH DE number 3925048