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The existence of symmetric cyclic Steiner quadruple systems

  • Hengrui Liu
  • Tao Feng
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Abstract

A Steiner quadruple system of order v, denoted by SQS(v), is called cyclic if it admits ℤv as a group of automorphisms acting regularly on its point set. A cyclic system is called symmetric cyclic, or S-cyclic, if the multiplier −1 preserves each block orbit individually. Grannell and Griggs conjectured in 1983 that such a system of order v exists if and only if v≡2,4,10,20pmod24 and every odd prime divisor p of v satisfies p≡1 or 5pmod12. This paper proves the conjecture. The main step is to construct an S-cyclic Steiner quadruple system of order 2p for every prime p≡1pmod4. Together with existing reduction theorems, these constructions determine which finite abelian groups admit a symmetric group-invariant Steiner quadruple system.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026existence,
  title = {{The existence of symmetric cyclic Steiner quadruple systems}},
  author = {Hengrui Liu and Tao Feng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25951},
  eprint = {2610.25951},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25951},
}
RIS
TY  - UNPB
TI  - The existence of symmetric cyclic Steiner quadruple systems
AU  - Hengrui Liu
AU  - Tao Feng
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.25951
UR  - https://arxiv.org/abs/2610.25951
AB  - A Steiner quadruple system of order v, denoted by SQS(v), is called cyclic if it admits ℤ_v as a group of automorphisms acting regularly on its point set. A cyclic system is called symmetric cyclic, or S-cyclic, if the multiplier −1 preserves each block orbit individually. Grannell and Griggs conjectured in 1983 that such a system of order v exists if and only if v≡2,4,10,20pmod24 and every odd prime divisor p of v satisfies p≡1 or 5pmod12. This paper proves the conjecture. The main step is to construct an S-cyclic Steiner quadruple system of order 2p for every prime p≡1pmod4. Together with existing reduction theorems, these constructions determine which finite abelian groups admit a symmetric group-invariant Steiner quadruple system.
ER  -