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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Polynomial identities in graded group rings, restricted Lie algebras and $ p$-adic analytic groups


Author: Aner Shalev
Journal: Trans. Amer. Math. Soc. 337 (1993), 451-462
MSC: Primary 16R99; Secondary 16S34, 16W50, 17B50, 20E18
MathSciNet review: 1093218
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Abstract: Let $ G$ be any finitely generated group, and let $ K$ be a field of characteristic $ p > 0$. It is shown that the graded group ring $ \operatorname{gr}(KG)$ satisfies a nontrivial polynomial identity if and only if the pro-$ p$ completion of $ G$ is $ p$-adic analytic, i.e. can be given the structure of a Lie group over the $ p$-adic field $ {\mathbb{Q}_p}$.

The proof applies theorems of Lazard, Quillen and Passman, as well as results on Engel Lie algebras and on dimension subgroups in positive characteristic.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1093218-X
PII: S 0002-9947(1993)1093218-X
Article copyright: © Copyright 1993 American Mathematical Society