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

 

The nilpotency class of finite groups of exponent $ p$


Author: Michael Vaughan-Lee
Journal: Trans. Amer. Math. Soc. 346 (1994), 617-640
MSC: Primary 20D15; Secondary 17B30
MathSciNet review: 1264152
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Abstract: We investigate the properties of Lie algebras of characteristic $ p$ which satisfy the Engel identity $ x{y^n} = 0$ for some $ n < p$. We establish a criterion which (when satisfied) implies that if $ a$ and $ b$ are elements of an Engel-$ n$ Lie algebra $ L$ then $ a{b^{n - 2}}$ generates a nilpotent ideal of $ L$. We show that this criterion is satisfied for $ n = 6,\,p = 7$, and we deduce that if $ G$ is a finite $ m$-generator group of exponent $ 7$ then $ G$ is nilpotent of class at most $ 51{m^8}$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1264152-8
PII: S 0002-9947(1994)1264152-8
Article copyright: © Copyright 1994 American Mathematical Society