The existence of pyramidal Steiner triple systems over abelian groups

A Steiner triple system STS(v) is called f-pyramidal if it has an automorphism group fixing f points and acting sharply transitively on the remaining v-f points. In this paper, we focus on the STSs that are f-pyramidal over some abelian group. Their existence has been settled only for the smallest admissible values of f, that is, f=0,1,3. In this paper, we complete this result and determine, for every f>3, the spectrum of values (f, v) for which there is an f-pyramidal STS(v) over an abelian group. This result is obtained by constructing difference families relative to a suitable partial spread.