Nilpotency of Some Lie Algebras Associated with p-Groups

Shumyatsky, Pavel

Canadian journal of mathematics, Tome 51 (1999) no. 3, pp. 658-672

Résumé

Let $L={{L}_{0}}+{{L}_{1}}$ be a ${{\mathbb{Z}}_{2}}$ -graded Lie algebra over a commutative ring with unity in which 2 is invertible. Suppose that ${{L}_{0}}$ is abelian and $L$ is generated by finitely many homogeneous elements ${{a}_{1}},.\,.\,.,{{a}_{k}}$ such that every commutator in ${{a}_{1}},.\,.\,.,{{a}_{k}}$ is ad-nilpotent. We prove that $L$ is nilpotent. This implies that any periodic residually finite ${2}'$ -group $G$ admitting an involutory automorphism $\phi $ with ${{C}_{G}}\left( \phi\right)$ abelian is locally finite.

Détail

DOI : 10.4153/CJM-1999-030-7

Mots-clés : 17B70, 20F50

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