Abstract 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 $$v-f$$ 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, $$f=0,1,3$$ f = 0 , 1 , 3 . In this paper, we complete this result and determine, for every $$f>3$$ 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.

Abstract Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner triple systems, little is known about their automorphism groups. In particular, there is no known congruence class representing the orders of a KTS with a number of automorphisms at least close to the number of points. We partially fill this gap by proving that whenever $$v \equiv 39$$ v≡39 (mod 72), or $$v \equiv 4^e48 + 3$$ v≡4e48+3 (mod $$4^e96$$ 4e96 ) and $$e \ge 0$$ e≥0 , there exists a KTS onvpoints having at least $$v-3$$ v-3 automorphisms. This is only one of the consequences of an investigation on the KTSs with an automorphism groupGacting sharply transitively on all but three points. Our methods are all constructive and yield KTSs which in many cases inherit some of the automorphisms ofG, thus increasing the total number of symmetries. To obtain these results it was necessary to introduce new types of difference families (the doubly disjoint ones) and difference matrices (the splittable ones) which we believe are interesting by themselves.