On flag‐transitive 2‐(k2,k,λ) $({k}^{2},k,\lambda )$ designs with λ∣k $\lambda | k$

Abstract

It is shown that, apart from the smallest Ree group, a flag-transitive automorphism group G of a 2-k k λ ( , , ) 2 design , with λ k  , is either an affine group or an almost simple classical group. Moreover, when G is the smallest Ree group,  is isomorphic either to the 2-(6 , 6, 2) 2 design or to one of the three 2-(6 , 6, 6) 2 designs constructed in this paper. All the four 2-designs have the 36 secants of a non-degenerate conic  of PG (8) 2 as a point set and 6-sets of secants in a remarkable configuration as a block set.

Analysing flag-transitive point-imprimitive 2-designs

“…In the following proposition, part (a) was proved in both [12, Propositions 6 and 8] and [13, Corollary 2.2 and Theorem 2.3]; and part (b) is proved in [12, Propositions 7 and 9] (and in both parts the block multiplicities may be greater than 1). In both papers these observations were applied to strengthen the results of [17] (see [12,Propositions 12 and 9] and [13,Theorem 2.4]). Note that although the applications in [12,13] are to symmetric designs, Proposition 2.8 is valid for all flag-transitive point-imprimitive designs.…”

“…In both papers these observations were applied to strengthen the results of [17] (see [12,Propositions 12 and 9] and [13,Theorem 2.4]). Note that although the applications in [12,13] are to symmetric designs, Proposition 2.8 is valid for all flag-transitive point-imprimitive designs.…”

“…In 2022, Mandić and Šubašić [12,Proposition 12] made small improvements in the bounds for symmetric designs in [17] (see also Proposition 2.8) and, building on classifications in [11,15,17] of the examples with λ ⩽ 4, they were able to identify all examples with parameter tuples in [17, Table 1] except for two parameter sequences. Even more recently, Montinaro [13,14] classified all flag-transitive, point-imprimitive symmetric designs for which k > λ(λ − 3)/2, and thereby ruled out these two parameter sequences. This completes the classification of flag-transitive, point-imprimitive symmetric designs with λ ⩽ 10 [14, Theorem 2.3]: there are eight such designs up to isomorphism (corresponding to four parameter sequences) all with λ ⩽ 4 and v < 100.…”

Sporadic simple groups as flag‐transitive automorphism groups of symmetric designs

“…In 2022, Mandić and Šubašić [12] treated all other open possibilities reported in [15] excluding two sets of parameters. Recently, Montinaro [13] made a contribution to the study of point‐imprimitive symmetric designs when $$k\gt \lambda (\lambda -3)\unicode{x02215}2$$ with a size of the intersection of block and imprimitivity class at least 3. In our study of point‐imprimitive symmetric designs with sporadic almost simple automorphism groups, we observe that the candidate parameter sets fall into type “a” or “b”, the first two parts in [15, Theorem 1.1].…”