On bounds for groups of multipliers of planar difference sets (Q1188159)

The author studies multiplier groups \(M\) of (not necessarily abelian) planar difference sets. Planar difference sets are in one-to-one correspondence with projective planes \(\Pi\) of order \(n\) admitting a sharply transitive automorphism group \(S\). A multiplier is a group automorphism of \(S\) which is also an automorphism of \(\Pi\). The author proves the following: (1) If \(M\) contains an involution, then \(n\) is a square and \(S=A\cdot B\) where \(A\) is an arc and \(B\) is a Baer subplane of \(\Pi\). (2) If \(M\) is abelian, then \(| M|\leq n+1\) or \(n\) is a square. (3) If \(M\) is abelian and \(S\) is cyclic, then \(| M|\leq n+1\) or \(n=4\). Moreover, planes of order 5 and 9 are characterized in terms of their multiplier groups. The proofs in the paper are group theoretic (and not number theoretic as quite often in connection with difference set problems).