$3$-pyramidal Steiner Triple Systems

TL;DR

This paper completely characterizes the existence of 3-pyramidal Steiner triple systems, identifying specific congruence conditions on the order v for their existence, thus solving a previously open problem for f=3.

Contribution

It provides a complete characterization of when 3-pyramidal Steiner triple systems exist based on the order v, filling a gap in the understanding of automorphism groups in combinatorial designs.

Findings

Existence of 3-pyramidal Steiner triple systems is characterized by specific modular conditions on v.

Such systems exist if and only if v ≡ 7, 9, 15 (mod 24) or v ≡ 3, 19 (mod 48).

The problem remains open for some values of v when f=1, but is fully solved for f=3.

Abstract

A design is said to be $f$-pyramidal when it has an automorphism group which fixes $f$ points and acts sharply transitively on all the others. The problem of establishing the set of values of $v$ for which there exists an $f$-pyramidal Steiner triple system of order $v$ has been deeply investigated in the case $f=1$ but it remains open for a special class of values of $v$. The same problem for the next possible $f$, which is $f=3$, is here completely solved: there exists a $3$-pyramidal Steiner triple system of order $v$ if and only if $v\equiv7,9,15$ (mod $24$) or $v\equiv 3, 19$ (mod 48).