Frattini theory for restricted Lie algebras
type: Item
instance of: scholarly article
title: Frattini theory for restricted Lie algebras
zbMATH Open document ID: 0578.17008
DOI: 10.1007/BF01195370
author: Mark Lincoln
author: David A. Towers
published in: Archiv der Mathematik
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 17B05
Mathematics Subject Classification ID: 17B30
zbMATH DE Number: 3925048
zbMATH Keywords: prime characteristic
zbMATH Keywords: Frattini ideal
zbMATH Keywords: nilpotent Lie algebras
zbMATH Keywords: Frattini subalgebra
zbMATH Keywords: restricted Lie algebras
zbMATH Keywords: nilradical
reviewed by: A. G. Gein
MaRDI profile type: Publication
cites work: On the cohomology of soluble Lie algebras
cites work: Q5732712
cites work: A Frattini Theory for Algebras
cites work: Two ideals of an algebra closely related to its Frattini ideal
description: scientific article; zbMATH DE number 3925048