Semiregular Relative Difference Sets in 2-Groups Containing a Cyclic Subgroup of Index 2

An ðm; u; k; lÞ relative difference set (RDS) in a group G relative to a subgroup U of order u and index m is a k-element subset R of G such that every element of g 2 G=U has exactly l representations g 1⁄4 r1r 1 2 with r1; r2 2 R and no identity element of U has such a representation. The subgroup U is often called the forbidden subgroup. If k 1⁄4 ul; then the RDS is called semiregular and its parameters are ðul; u; ul; lÞ: If an abelian p-group contains a semiregular RDS, then its exponent is relatively low (see [3, 8, 9, 11, 12]). However, this is not true for non-abelian p-groups. For each integer n53; the generalized quaternion group Q2n of order 2 and exponent 2 1 has a ð2 ; 2; 2 ; 2 Þ RDS and for an odd prime p; the modular p-group M3ðpÞ of order p and exponent p has a ðp; p; p; pÞ RDS (see [5]). In [5] semiregular RDSs in non-abelian 2-groups of maximal exponent with the condition that the forbidden subgroup is normal (see Result 2.3 in Section 2) were studied. The conclusion was that there are no semiregular RDSs in the groups except when G 1⁄4 Q2n or M4ð2Þ: In the present article, the condition on normality of the forbidden subgroup is removed and the following is shown: