A note about additive \(2\)-\((v,5,\lambda)\) designs (Q2906241)

Let \(G\) be a finite abelian group, and let \(3\leq k\leq \left| G\right|\). Define the incidence structure \(\mathcal{D}_k(G)\) as follows: the points of \(\mathcal{D}_k(G)\) are the elements of \(G\), and the blocks of \(\mathcal{D}_k(G)\) are those \(k\)-subsets \(\{c_1,c_2,\hdots,c_k\}\) of \(G\) that have the property that \(c_1+c_2+\cdots+c_k=0\). A question that arose in previous research by the author and collaborators is under which conditions on \(G\) and \(k\) the incidence structure \(\mathcal{D}_k(G)\) is a \(2\)-design. Answers to this question had previously been given for \(k=3\) and \(k=4\) (\(G\) must be an elementary abelian \(3\)-group, respectively \(2\)-group). It also had been shown earlier that \(\mathcal{D}_k(G)\) is a \(2\)-design if \(G\) is an elementary abelian \(p\)-group where \(p\) divides \(k\). In the article under review the author obtains a necessary result for \(\mathcal{D}_5(G)\) to be a \(2\)-design. The author shows that if \(\mathcal{D}_5(G)\) is a \(2\)-design, then necessarily \(G\) has odd order. The proof is set up as follows: the abelian groups of even order are divided into five classes, and for each of these classes it is shown that the resulting incidence structure \(\mathcal{D}_5(G)\) cannot be a \(2\)-design. The arguments involved are mostly counting arguments combined with some group theory.