On automorphism groups of divisible designs: II: Group invariant generalized conference matrices (Q1112043)

We prove the non-existence of conference matrices which are invariant under a regular Abelian group. In the non-Abelian case, a group invariant conference matrix of order \(2\lambda +2\) can only exist if \(2\lambda +1\) is a perfect square; no examples of this situation are known, however. These results are applied to obtain non-existence theorems for group invariant generalized conference matrices and thus for splitting relative difference sets. In particular, no splitting Abelian affine difference set of odd order s can exist; for \(s\equiv 3\) mod 4, the non-Abelian case is likewise impossible.