On \((p^ a,p,p^ a,p^{a-1})\)-relative difference sets
Let \(G\) be an abelian group of order \(m \cdot n\) containing a subgroup \(N\) of order \(n\). An (abelian) relative difference set \(D\) with parameters \((m,n,k, \lambda)\) is a \(k\)-subset of \(G\) such that the list of differences \((d - d' : d, d' \in D)\) covers all the elements in \(G \backslash N\) exactly \(\lambda\) times, but no non-zero element in \(N\). Relative difference sets with \(m = k = n \lambda\) (the so-called semiregular case) are of particular interest in view of their connection to projective planes via planar functions, see \textit{A. Pott} [Finite geometry and character theory, Lect. Notes Math. 1601 (1995; Zbl 0818.05001)]. The paper under review contains necessary and sufficient conditions for the existence of semiregular relative difference sets in \(p\)-groups where \(n = p\). Relative \((p^{2a}, p, p^{2a}, p^{2a - 1})\)-difference sets exist if and only if the exponent of \(G\) is at most \(p^{a + 1}\). Similarly, relative \((p^{2a + 1}, p, p^{2a + 1}, p^{2a})\)-difference sets with \(p = 2\) exist if and only if the exponent of \(G\) is at most \(2^{a + 2}\). The paper contains results for the case \(p\) odd, too. These results have been improved recently by J. Davis and J. Jedwab.
scientific article; zbMATH DE number 777669
scientific article