Singer groups of biplanes of order 25 (Q1123196)

We prove that biplanes of order 25 (i.e. a 2-(352,27,2) symmetric design whose existence is still open) cannot admit certain types of Singer groups. These groups include all the Abelian groups of order 352 and certain non-Abelian groups. These results, in particular, correct erroneous proofs of Lander for the nonexistence of (352,27,2) difference sets in \(Z_ 2\times Z_{16}\times Z_{11}\) and \(Z_ 4\times Z_ 8\times Z_{11}\). Arasu, Davis, Jungnickel and Pott corrected the cases where the underlying group is either \(Z_{11}\times Z_ 8\times Z_ 2\times Z_ 2\) or \(Z_{11}\times Z_ 4\times Z_ 4\times Z_ 2\).