Planes and Biplanes

Biplanes with flag-transitive automorphism groups of almost simple type, with exceptional socle of Lie type

“…For k = 3, 4, and 5 the biplanes are unique up to isomorphism [5], for k = 6 there are exactly three non-isomorphic biplanes [11], for k = 9 there are exactly four nonisomorphic biplanes [31], for k = 11 there are five known biplanes [3,9,10], and for k = 13 there are two known biplanes [1], in this case, it is a biplane and its dual.…”

Biplanes with flag-transitive automorphism groups of almost simple type, with classical socle

“…Since k divides 2(|G x |, v − 1), we have k 2 < v in all cases except in the case L 2 (7) < U 3 (3). In this last case v = 36, but then there is no k such that k(k − 1) = 2(v − 1), which is a contradiction.…”

“…If q ≤ 3 then let W = u 1 , u 2 , u 3 . Taking g ∈ G \ G x acting trivially on W ⊥ we see that now k divides n(n−1)(n−2)(q+1) 3 …”

“…For k = 3, 4, and 5 the biplanes are unique up to isomorphism [6], for k = 6 there are exactly three non-isomorphic biplanes [13], for k = 9 there are exactly four nonisomorphic biplanes [26], for k = 11 there are five known biplanes [3,10,11], and for k = 13 there are two known biplanes [1], in this case, it is a biplane and its dual.…”

On primitivity and reduction for flag-transitive symmetric designs

“…By the above inequality, this list does not include, for example, the parameters (15,7,3) for which five non-isomorphic designs exist [6, p. 11].…”