2-minimal subgroups in finite exceptional groups of Lie type

Abstract

Suppose G is a finite group, S a Sylow 2-subgroup of G and $B=N_G\left(S\right)$. Then a subgroup P of G is a 2-minimal subgroup of G with respect to B if and only if B is contained in a unique maximal subgroup of P. Here the 2-minimal subgroups of the finite simple groups of exceptional Lie type are classified. This classification yields detailed descriptions of the 2-minimal subgroups and, by way of illustration, we list explicitly the 2-minimal subgroups for $G_2\left(3\right)$, $F_4\left(3\right)$, $E_6\left(19\right)$, $E_7\left(5^3\right)$ and $E_8\left(11\right)$.