## A new maximal subgroup of $E_8$ in characteristic $3$

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- by David A. Craven, David I. Stewart and Adam R. Thomas PDF
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## Abstract:

We prove the existence and uniqueness up to conjugacy of a new maximal subgroup of the algebraic group of type $E_8$ in characteristic $3$. This has type $F_4$, and was missing from previous lists of maximal subgroups produced by Seitz and Liebeck–Seitz. We also prove a result about the finite group $H={}^3\!D_4(2)$, namely that if $H$ embeds in $E_8$ (in any characteristic $p$) and has two composition factors on the adjoint module then $p=3$ and $H$ lies in a conjugate of this new maximal $F_4$ subgroup.## References

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## Additional Information

**David A. Craven**- Affiliation: School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom
- MR Author ID: 833948
- Email: d.a.craven@bham.ac.uk
**David I. Stewart**- Affiliation: School of Mathematics, Statistics and Physics, Herschel Building, Newcastle University, Newcastle NE1 7RU, United Kingdom
- MR Author ID: 884527
- Email: david.stewart@ncl.ac.uk
**Adam R. Thomas**- Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom
- MR Author ID: 1091953
- Email: Adam.R.Thomas@warwick.ac.uk
- Received by editor(s): April 22, 2021
- Received by editor(s) in revised form: June 28, 2021
- Published electronically: January 20, 2022
- Additional Notes: The first author was supported by the Royal Society during the course of this research
- Communicated by: Martin Liebeck
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**150**(2022), 1435-1448 - MSC (2020): Primary 20G41
- DOI: https://doi.org/10.1090/proc/15759
- MathSciNet review: 4375734