Nilpotency of Some Lie Algebras Associated with $p$-Groups

http://dx.doi.org/10.4153/CJM-1999-030-7
Canad. J. Math. 51(1999), 658-672

  Published:1999-06-01
 Printed: Jun 1999

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 $a_1,\dots,a_k$ such that every commutator in $a_1,\dots,a_k$ 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 $C_G(\phi)$ abelian is locally finite.