The existence of pyramidal Steiner triple systems over abelian groups

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 $$v-f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo>-</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> 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, $$f=0,1,3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> . In this paper, we complete this result and determine, for every $$f&gt;3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , 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.