As a consequence of the classification of the finite simple groups, it has
been possible in recent years to characterize Steiner t-designs, that is t -(v, k, 1) designs,
mainly for t = 2, admitting groups of automorphisms with sufficiently strong
symmetry properties. However, despite the finite simple group classification, for
Steiner t-designs with t > 2 most of these characterizations have remained longstanding
challenging problems. Especially, the determination of all flag-transitive
Steiner t-designs with 3 ≤ t ≤ 6 is of particular interest and has been open for about
40 years (cf. Delandtsheer (Geom. Dedicata 41, p. 147, 1992 and Handbook of Incidence
Geometry, Elsevier Science, Amsterdam, 1995, p. 273), but presumably dating
back to 1965).
The present paper continues the author’s work (see Huber (J. Comb. Theory Ser.
A 94, 180–190, 2001; Adv. Geom. 5, 195–221, 2005; J. Algebr. Comb., 2007, to
appear)) of classifying all flag-transitive Steiner 3-designs and 4-designs. We give a
complete classification of all flag-transitive Steiner 5-designs and prove furthermore
that there are no non-trivial flag-transitive Steiner 6-designs. Both results rely on the
classification of the finite 3-homogeneous permutation groups. Moreover, we survey
some of the most general results on highly symmetric Steiner t-designs.