Suzuki groups and 2-\((v,k,1)\) designs (Q5942873)

Let \(G\) be a block-primitive group of automorphisms of a Steiner system \(D=S(2,k,v)\). If \(G\) contains the Suzuki group \(S=\text{Sz}(2^{2m+1})\) and it is contained in \(\text{ Aut}(S)\), then \(G\) is point-primitive. In any case the socle of \(G\) is not an \(\text{Sz}(q)\). This supports Doyen-Deleandsheer's conjecture that a block-primitive group of automorphisms of a Steiner system is also point-primitive.