On Singular 1-Rotational Steiner 2-Designs

A Steiner 2-design is 1-rotational over a groupGif it admitsGas an automorphism group fixing one point and acting regularly on the remainder. 1-rotational Steiner 2-designs have come into fashion since 1981, when Phelps and Rosa (Discrete Math.33(1981), 57–66) studied Steiner triple systems that are 1-rotational over the cyclic group. While all 1-rotational Steiner 2-designs con- structed in the past have exactly oneshort block-orbit, in this paper we also consider 1-rotational Steiner 2-designs not having this property. We call themsingularand we show that they are quite rare. In particular, we enumerate all the abelian 1-rotational 2-(49, 4, 1) designs.