Identity with constants in a Chevalley group of type 𝐹₄

Nesterov, V.; Stepanov, A.

doi:10.1090/S1061-0022-2010-01119-1

Identity with constants in a Chevalley group of type ${\mathrm F}_4$
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by V. Nesterov and A. Stepanov
Translated by: the authors

St. Petersburg Math. J. 21 (2010), 819-823

DOI: https://doi.org/10.1090/S1061-0022-2010-01119-1

Published electronically: July 15, 2010

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Abstract:

N. L. Gordeev proved that a generalized group identity holds in Chevalley groups with multiply laced root systems. It was also shown that a stronger identity is valid for the Chevalley groups of types $\mathrm {B}_l$ and $\mathrm {C}_l$. In the present paper, it is proved that this strong identity is fulfilled in Chevalley groups of type $\mathrm {F}_4$ and fails to be true in Chevalley groups of type $\mathrm {G}_2$. The main result of the paper is the last ingredient in the proof of the claim that the lattice of intermediate subgroups between $G(\mathrm {F}_4,R)$ and $G(\mathrm {F}_4,A)$ is standard for an arbitrary pair of rings $R\subseteq A$ with 2 invertible.

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