The existence of symmetric cyclic Steiner quadruple systems

A Steiner quadruple system of order $v$, denoted by $\mathrm{SQS}(v)$, 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 $v\equiv 2,4,10,20(\mathrm{m}\mathrm{o}\mathrm{d}24)$ and every odd prime divisor $p$ of $v$ satisfies $p\equiv 1$ or $5(\mathrm{m}\mathrm{o}\mathrm{d}12)$. This paper proves the conjecture. The main step is to construct an $S$-cyclic Steiner quadruple system of order $2p$ for every prime $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$. Together with existing reduction theorems, these constructions determine which finite abelian groups admit a symmetric group-invariant Steiner quadruple system.