Finite exceptional groups of Lie type and symmetric designs

On Flag-Transitive $2$-$(k^{2}, k, λ)$ Designs with $λ\mid k$

“…Since then a special attention was given to the case λ > 1. A classification of the flag-transitive 2-designs with gcd(r, λ) = 1, λ > 1 and G AΓL 1 (q), where r is the replication number of D, has been announced by Alavi, Biliotti, Daneshkakh, Montinaro, Zhou and their collaborators in [2] and proven in [3], [4], [5], [8], [10], [11], [30], [37], [39], [40], [41], [42], [44], [45] and [46]. Moreover, recently the flag-transitive 2-designs with λ = 2 have been investigated by Devillers, Liang, Praeger and Xia in [19], where it is shown that apart from the two known symmetric 2-(16, 6, 2) designs, G is primitive of affine or almost simple type.…”

“…Actually, (3) and ( 4) are special cases of (2), since 2 G 3 (3) ∼ = P ΓL 2 (8). It worth noting that the example in (3) is not contained in [19] and hence it is presumably new.…”

“…Then we combine some group theoretical arguments, in particular those developed in [17], together with some numerical properties of the binomial coefficients to exclude the case. Finally, when G is an exceptional group of Lie type, the reduction to 2 G 2 (3) in its permutation representation of degree 36 is settled by transferring the arguments developed in [3] and in [8] to our context. The key point of the analysis of the 2-designs admitting 2 G 2 (3) as a flag transitive automorphism group is to see that 2 G 2 (3) acts on P G 2 (8) preserving a nondegenerate conic C, since 2 G 2 (3) ∼ = P ΓL 2 (8).…”

“…Finally, when G is an exceptional group of Lie type, the reduction to 2 G 2 (3) in its permutation representation of degree 36 is settled by transferring the arguments developed in [3] and in [8] to our context. The key point of the analysis of the 2-designs admitting 2 G 2 (3) as a flag transitive automorphism group is to see that 2 G 2 (3) acts on P G 2 (8) preserving a nondegenerate conic C, since 2 G 2 (3) ∼ = P ΓL 2 (8). Hence, its permutation representation of degree 36 is equivalent to that on the set of secants to C. Some geometry of P G 2 (8) is then used to complete the proof of the case.…”

“…Suppose that G x is not parabolic. Then G x is one of the groups listed in [8], Theorem 1.6, or equivalently in Tables 2 and 3 of [3], since G x is a large maximal subgroup of G by Lemma 2.2(2). In [3], Alavi points out that the first and the second column of Table 2 contains X and X x , respectively, the third one contains a lower bound ℓ v for v = [X : X x ], and the fourth one contains an upper bound u r for r, determined by using the fact that r is a common divisor of v − 1, of |G x | and of the subdegrees of G x by his Lemma 4.…”