A sharp exponent bound for McFarland difference sets with \(p=2\)
A McFarland difference set is a difference set with \(v=q^{d+1}[1+(q^{d+1}-1)/(q-1)]\), \(k=q^d(q^{d+1}-1)/(q-1)\), \(n=q^{2d}\) (\(q=p^f\), \(p\) prime). If there exists a such difference set in an abelian group \(G\) of exponent \(m=p^\alpha m'\) (with \((m',p)=1\)) and there exists an integer \(j\) with \(p^j\equiv -1 \pmod{m'}\) then for \(p\) odd the \(p\)-Sylow group \(P\) of \(G\) is elementary abelian and for \(p=2\) and \(f\geq 2\) the exponent of \(P\) is less than \(5\). The proof uses the authors' results in [The structure of abelian groups containing McFarland difference sets, J. Comb. Theory, Ser A 70, No. 2, 313-322 (1995; Zbl 0830.05013)], and in passing a proof of a result in a previous paper is fixed. Similar results are promised for relative difference sets.
scientific article; zbMATH DE number 1093109