Symmetric designs admitting flag-transitive and point-primitive automorphism groups associated to two dimensional projective special groups

On flag-transitive automorphism groups of symmetric designs

“…Since also (t − 4) 2 + 16 4 3 t, we conclude that 2 t 71, and so (3.1) and (3.2) imply that k = t(t 2 y 2 − 8ty + 16y + 16) 64 and λ = (ty − 4) 2 + 16y 64 , where 2 t 71 and 1 y 5. For these values of t and y, considering the fact that m 5, k λ 2 and λ divides k, we obtain (v, k, λ) = (121, 25,5) or (441, 56, 7) respectively when (t, y) = (5,4) or (8,3). These possibilities can be ruled out by [4] Proof.…”

“…In Table 1, we list some small examples of such designs with λ 3. To our knowledge the design in Line 2 is the only point-primitive example of symmetric designs with v 2500 satisfying the conditions of Theorem 1.1 and this motivates the authors to investigate symmetric designs admitting symplectic automorphism groups [3]. More examples of symmetric designs admitting flagtransitive and point-imprimitive automorphism groups can be found in [20] and references therein.…”

Almost Simple Groups of Lie Type and Symmetric Designs with $\lambda$ Prime

“…In order to prove Theorem 1 in Section 4, as noted above, by [6,Corollary 1.2], we only need to consider the case where the socle X of G is a finite simple classical group. In particular, by [1,3,5,7,15], in the case where X is a linear or unitary group, we can assume that the dimension of the underlying vector space is at least 5. Moreover, we include all possible symmetric (v, k, λ) designs for λ = 2, 3 obtained in [18,30,32] and therein references, and so we can also assume that λ 5.…”

“…In this section, we provide some examples of symmetric designs with λ prime admitting a flag-transitive automorphism almost simple group with socle X. We remark here that the designs in Table 1 can be found in [3,7], but the construction given here is obtained by GAP [19].…”

“…We are now interested in possible classification of symmetric (v, k, λ) designs D with λ prime admitting a flag-transitive and point-primitive almost simple automorphism group G. We have already shown in [6] that almost simple exceptional groups of Lie type give rise to no possible symmetric designs with λ prime. In the present paper, we focus on the case where G is an almost simple group with socle X being a finite simple classical group of Lie type, and prove that D is either the point-hyperplane design of a projective space PG n−1 (q), or it is of parameters (7,4,2), (11,5,2), (11,6,3) or (45,12,3), and we give detailed information of these designs in Section 2.…”