Steiner quadruple systems with point-regular abelian automorphism groups

In this article we present a graph theoretic construction of Steiner quadruple systems (SQS) admitting Abelian groups as point-regular automorphism groups. The resulting SQS has an extra property that we call A-reversibility, where A is the underlying Abelian group. In particular, when A is a 2-group of exponent at most 4, it is shown that an A-reversible SQS always exists. When the Sylow 2-subgroup of A is cyclic, we give a necessary and sufficient condition for the existence of an A-reversible SQS, which is a generalization of a necessary and sufficient condition for the existence of a dihedral SQS by Piotrowski (1985). This enables one to construct A-reversible SQS for any Abelian group A of order v such that for every prime divisor p of v there exists a dihedral SQS(2p).