A new existence proof of Fi22 in 2E6(2), a computational approach

Abstract

The purpose of this article is to give a new, explicit and elementary construction of the $F_{i_{22}}$ in ${}^2{E}_6(2),$ using the notion of M-sets introduced by the second author, and properties of the generalized quadrangle $(\mathbb{P},\mathcal{L})$ of type $O_6^{-}(2).$ This construction is elementary and explicit as the transpositions generating $F_{i_{22}}$ in ${}^2{E}_6(2)$ have been explicitly constructed and implemented in GAP, which might be very helpful and well suited to people who do computations with this group. It is remarkable to mention that this work supports the work of Cuypers et al., where the existence of $F_{i_{22}}$ has been assumed then the embedding of the sporadic simple group $F_{i_{22}}$ in the group ${}^2{E}_6(2)$ has been given. In fact our approach for the construction is completely different from Fischer’s construction.

Keywords:

• Chevalley group
• generalized quadrangle
• M-sets
• root elements
• 3-transportation group
• unitary group

MATHEMATICS SUBJECT CLASSIFICATION:

• 17A75
• 17A45

Acknowledgment

The authors would like to thank Kuwait foundation for the advancement of sciences for supporting this project No. PR17-165M-07, Mr. H. J. Schaeffer and Prof. C. Hering for their remarks and valuable suggestions on this article, and to Dr. B. Al-Hasanat for his neat typing of this article.