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arXiv · 2610.25951

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 $\operatorname{SQS}(v)$, is called cyclic if it admits $\mathbb Z_v$ 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 $v\equiv2,4,10,20\pmod{24}$ and every odd prime divisor $p$ of $v$ satisfies $p\equiv1$ or $5\pmod{12}$. This paper proves the conjecture. The main step is to construct an $S$-cyclic Steiner quadruple system of order $2p$ for every prime $p\equiv1\pmod4$. Together with existing reduction theorems, these constructions determine which finite abelian groups admit a symmetric group-invariant Steiner quadruple system.

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Hengrui Liu, Tao Feng. 2026-10-19. The existence of symmetric cyclic Steiner quadruple systems. https://arxiv.org/abs/2610.25951

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