Radical 3-subgroups of F4(q) with q even

The inductive blockwise Alperin weight condition for the Chevalley groups $F_4(q)$

“…The results for the principal 3-block reveal a distinctive different behaviour in the cases when 9 divides or does not divide q 2 − 1. Our investigations for ℓ = 3 are largely supported by the fact that 3 is good for all proper Levi subgroups of G, and that the radical 3-subgroups of G, which are the candidates for the weight subgroups have been classified in [6] and [4]. The missing classification of the radical 2-subgroups of G prevents us from extending our results to the case ℓ = 2.…”

“…. , n 4 satisfy the relations exhibited in [77, 4.6, Equations ( 1)-( 4)], and that Ŵ has order 2 4 |W |. Thus if p is odd, Ŵ is indeed the extended Weyl group as defined in [77,Définition 2.2].…”

“…Section 6 is dedicated solely to the prime 3. We recall the construction of the radical 3-subgroups of G established in [4] and [6] to some detail, as this will be important later on. The defect groups of the 3-blocks of G are, in particular, radical 3-subgroups.…”

“…or the G-class type of s is one of (1, 1), (4, 1), (4, 2), (14, 1), (14,4), (15,1), (15,3) or (15,5). Moreover, the following statements hold.…”

“…We can also assume that C G (s) is not a torus. As σ 1 interchanges the long root subgroups of G with its short root subgroups, the tables in [55] only leave the possibilities (1, 1), (4, k), (14, k) or (15, k) for the G-class types of s. As σ interchanges the maximal tori of type (20,12) and (20,17), it also swaps the classes of type (14, 2) and (14, 3) and the classes of type (15, 2) and (15,4). Let us now prove the remaining statements.…”