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

Open Access article
 Printed: Jun 1999
  • Pavel Shumyatsky
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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.
MSC Classifications: 17B70, 20F50 show english descriptions Graded Lie (super)algebras
Periodic groups; locally finite groups
17B70 - Graded Lie (super)algebras
20F50 - Periodic groups; locally finite groups

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