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In this paper we present a graph theoretic construction of Steiner quadruplesystems (SQS) admitting abelian groups as point-regular automorphism groups.The resulting SQS has an extra property which we call A-reversibility, where Ais the underlying abelian group. In particular, when A is a 2-group of exponentat most 4, it is shown that an A-reversible SQS always exists. When the Sylow2-subgroup of A is cyclic, we give a necessary and sufficient condition for theexistence of an A-reversible SQS, which is a generalization of a necessary andsufficient 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 orderv such that for every prime divisor p of v there exists a dihedral SQS(2p).

Steiner quadruple systems with point-regular abelian automorphism groups