On singular 1-rotational Steiner 2-designs

A Steiner 2-design is 1-rotational over a group G if it admits G as an auto morphism group fixing one point and acting regularly on the remainder, 1-ro tational Steiner 2-designs have come into fashion since 1981, when Phelps a nd Rosa (Discrete Math. 33 (1981), 57-66) studied Steiner triple systems th at are 1-rotational over the cyclic group. While all I-rotational Steiner 2 -designs constructed in the past have exactly one short block-orbit, ill th is paper we also consider I-rotational Steiner 2-designs not having this pr operty. We call them singular and we show that they are quite rare. In part icular, we enumerate all the abelian 1-rotational 2-(49, 4, 1) designs. (C) 1999 Academic Press.