Research on the relation between relative difference sets in non-abelian groups and group characters

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In this research we studied the relation between semi-regular relative difference sets (RDS) and group characters. Our main results are as follows :Theorem 1. Let G be a group with an abelian normal subgroup A of index 2 inverted by an element t ∈ G＼A of order 2. Assume that G has a semi-regular RDS, say R, relative to a normal subgroup U of G.Then a Sylow p-subgroup of A is non-cyclic for any prime p dividing u.J.A.Davis has constructed examples of semi-regular RDS in Q_<2^m>. In general we obtained the following.Theorem 2. Let R be a semi-regular RDS in a dicyclic group Q_<4m> relative to a normal subgroup U.If |U| is even, then |U|=2.Concerning dihedral groups D_<2m> and semi-dihedral groups SD_<2^m>, we have the following result.Theorem 3 There is no semi-regular RDS in any dihedral group.As a conclusion in this research we have shown the following.Theorem 4 Let G be a non-abelian p-group with a maximal cyclic subgroup. If G contains a semi-regular RDS relative to a normal subgroup U, then one of the following holds :(i) G【similar or equal】Q_<2^n> and U【similar or equal】Z_2,(ii) G【similar or equal】M_n (p) and U【similar or equal】Z_p with p an odd prime,(iii) G【similar or equal】M_4 (2) and U【similar or equal】Z_2×Z_2.