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 ≡ 7, 9, 15 (mod 24) or v ≡ 3, 19 (mod 48).

Highlights

  • 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

  • A Steiner triple system of order v, briefly STS(v), is a pair (V, B) where V is a set of v points and B is a set of 3-subsets of V with the property that any two distinct points are contained in exactly one block

  • Steiner triple systems having an automorphism with a prescribed property or an automorphism group with a prescribed action have drawn much attention since a long time

Read more

Summary

Introduction

A Steiner triple system of order v, briefly STS(v), is a pair (V, B) where V is a set of v points and B is a set of 3-subsets (blocks or triples) of V with the property that any two distinct points are contained in exactly one block. We want to consider the problem of determining the set of values of v for which there exists a STS(v) with an automorphism group fixing f points and acting sharply transitively on the other v − f points. Such a STS will be called f -pyramidal. It is natural to study the possible case that is f = 3 This is because, as we are going to see in the lemma, the fixed points of an f -pyramidal STS(v) form a subsystem of order f so that, for f = 0, we have f ≡ 1 or 3 (mod 6); if f = v, f < v/2. We have to translate our problem into algebraic terms: any f -pyramidal STS is completely equivalent to a suitable difference family

Difference families and pyramidal STSs
The “if part”
The “only if part”
Findings
Conclusion
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call