On quasiregular collineation groups of projective planes (Q1179520)

The authors investigate quasiregular collineation groups \(\Gamma\) of type (d) in the Dembowski-Piper classification [\textit{P. Dembowski} and \textit{F. Piper}, Math. Z. 99, 53--75 (1967; Zbl 0145.41003)]. They assume \(\Gamma\) to be Abelian, and they prove that the Sylow 2-subgroup of \(\Gamma\) as well as the Sylow 2-subgroup of its multiplier group have to be cyclic. The results are then used to obtain conditions on the existence of affine difference sets: a theorem about multipliers of even order, a study of the geometric properties of multipliers, and finally some non-existence tests derived from these results.