The nilpotency class of finite groups of exponent 𝑝

Vaughan-Lee, Michael

doi:10.1090/S0002-9947-1994-1264152-8

The nilpotency class of finite groups of exponent $p$
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Trans. Amer. Math. Soc. 346 (1994), 617-640 Request permission

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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