Flag-transitive Steiner designs

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The study of combinatorial or geometric structures through their groups of automorphisms has gained significant interest, seen as a natural extension of Felix Klein’s Erlangen program from 1872. This field has practical applications in design theory, coding theory, and cryptography, particularly for finite structures. This research monograph focuses on the classification of flag-transitive Steiner designs, which are combinatorial t-(v, k, 1) designs with automorphism groups acting transitively on incident point-block pairs. Recent advancements, particularly due to the classification of finite simple groups, have enabled the characterization of Steiner t-designs, especially for t=2. Notable achievements include W. M. Kantor's classification of all point 2-transitive Steiner 2-designs in 1985 and the near-complete determination of flag-transitive Steiner 2-designs by a group of researchers in 1990. However, despite these developments, the characterization of Steiner t-designs with t > 2 remains a challenging area, particularly the determination of all flag-transitive Steiner t-designs for 3 ≤ t ≤ 6, which has been a focus of research for over 40 years.