2-Minimal Subgroups in Finite Exceptional Groups of Lie Type

Suppose G is a finite group, S a Sylow 2-subgroup of G and B = N_G(S). 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 G₂(3), F₄(3), E₆(19), E₇(5 3 ) and E₈(11).