| A Steiner quadruple system of order v(briefly an SQS(v)) is an ordered pair(X, B)where X is a set of cardinality v whose elements are called points, and B is a set 4-subsets of X, called blocks, with the property that every 3-subset of X is contained in a unique block. An automorphism α of an SQS(X, B) is a permutation of X leaving B invariant,i.e., {{α(x) : x ∈ B} : B ∈ B} = B. All automorphisms of an SQS form a group, called the full automorphism group of the SQS. Any subgroup of the full automorphism group is called an automorphism group of the SQS. In 1992, Hartman and Phelps surveyed the progress on SQSs having automorphism groups, including cyclic SQSs, rotational SQSs and dihedral SQSs(the dihedral group Dv/2is an automorphism group of SQS(v)). Since then, a great deal of research has been done on cyclic SQSs and rotational SQSs, but no systematic study has been done on dihedral SQSs. The known results about dihedral SQSs include the construction of dihedral SQS(2p) given by Hartman(1980) where p ≡ 7(mod 12) is a prime and the dihedral SQSs from S-cyclic SQSs. The existence problem of dihedral SQSs was posed by Hartman and Phelps in their survey paper in 1992. In this thesis, we shall systematically research constructions of dihedral SQSs.Dihedral SQSs can be constructed from dihedral G designs, and G designs can be obtained from H designs. Firstly, we study constructions of dihedral H designs, including the constructions of a dihedral H(p, 2, 4, 3) from a special rotational SQS(p+1), a dihedral H(m, 2, 4, 3) from a cyclic SQS(m) and a recursive construction of dihedral H designs by means of symmetric semi-cyclic H designs. And we obtain some infinite families of dihedral H designs by using these constructions. Then, we give a construction of dihedral G designs from dihedral H designs by using the one-factorizations of complete graphs with dihedral groups and a recursive construction of special dihedral G designs. At last,some new infinite families of dihedral SQSs are obtained by using the results of rotational SQSs, cyclic SQSs etc. and recursive constructions. |
