Flag-transitive, point-imprimitive \(2\)-\((v,k,\lambda )\) symmetric designs with \(k\) and \(\lambda\) prime powers (Q2034990)

A symmetric 2-\((v, k, \lambda)\) design \(D\) is an incidence structure \((P, B)\), where \(P\) is a set of \(v\) points and \(B\) is a set of \(k\)-subsets of \(P\) called blocks with the property that for any two distinct points there are exactly \(\lambda\) blocks containing them. Let \(k\) and \(\lambda\) be prime powers. If \(D\) admits a flag-transitive, point-imprimitive automorphism group \(G\), the authors establish that both \(k\) and \(\lambda\) must be powers of 2. Moreover, there exists an integer \(m\) such that either (a) \(D\) has parameters \((v, k, \lambda)\) \(= (2^{2m+2}-1, 2^{2m+1}, 2^{2m})\), and \(G\) preserves a partition of the points into \(2^{m+1} +1\) classes of size \(2^{m+1} -1\), or (b) \(D\) has parameters \((v, k, \lambda) = ((2^{2m-1} -2^m + 1)(2^{m-1} + 1), 2^{2m-1}, 2^m)\), and \(G\) preserves a partition of the points into \(2^{2m-1}-2^m + 1\) classes of size \(2^{m-1} + 1\).