GBRDs with block size three over 2-groups, semi-dihedral groups and nilpotent groups
A generalized Bhaskar Rao design \(\mathrm{GBRD}(v, k, \lambda; G)\) is a \(v \times b\) array (\(b = \frac{\lambda v(v-1)}{k(k-1)}\)), each entry of which is either \(0\) or an element of \(G\) such that each row has \(r = \frac{\lambda(v-1)}{k-1}\) entries from \(G\) and each column has \(k\) entries from \(G\), and for each pair of distinct rows \((x_1, x_2, \dots, x_b)\) and \((y_1, y_2,\dots, y_b)\) the list \(x_iy_i^{-1}\): \(i\in\{1,2,\dots,b\}\) with \(x_i,y_i \in G\), contains each group element exactly \(\lambda/|G|\) times. Well-known necessary condition for a \(\mathrm{GBRD}(v, 3,\lambda;G)\) to exist are {\parindent=7mm \begin{itemize}\item[(i)]\(\lambda\equiv 0\bmod |G|\); \item[(ii)]\(\lambda(v-1)\equiv0\bmod 2\); \item[(iii)]\(\lambda v(v-1)\equiv 0\bmod 6\) when \(|G|\) is odd, and \(0\bmod{24}\) when \(|G|\) is even; and \item[(iv)]If \(v = 3\), and \(G\) has a non-trivial cyclic Sylow \(2\)-subgroup, then \(\lambda\equiv0\bmod{2|G|}\). \end{itemize}} It was conjectured that these conditions are sufficient as long as \(v>3\), and it was verified for many groups. The current paper establishes the conjecture for nilpotent groups of even order, as well as for semi-dihedral groups. (Note that the conjecture holds for nilpotent groups of odd order and was established in 1992 by \textit{W. D. Palmer} [``A composition theorem for generalized Bhaskar Rao designs,'' Australas. J. Comb. 6, 221--228 (1992; Zbl 0776.05010)]).
scientific article; zbMATH DE number 5852482