GBRDs with block size three over 2-groups, semi-dihedral groups and nilpotent groups
type: Item
instance of: scholarly article
title: GBRDs with block size three over 2-groups, semi-dihedral groups and nilpotent groups
zbMATH Open document ID: 1233.05042
author: R. Julian R. Abel
author: D. Combe
author: Adrian M. Nelson
author: W. D. Palmer
published in: The Electronic Journal of Combinatorics
publication date: 2011-02-17T00:00:00Z
full work available at URL: https://eudml.org/doc/223637
full work available at URL: v18i1p32.html
review text: 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)]).
reviewed by: Wen-Fong Ke
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 5852482
zbMATH Keywords: generalized Bhaskar Rao design (GBRD)
zbMATH Keywords: nilpotent group
zbMATH Keywords: semi-dihedral group
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5852482