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Keywords:
Conformal mapping; geodesic mapping; conformal-geodesic mapping; initial conditions; (pseudo-) Riemannian space
Conformal mapping; geodesic mapping; conformal-geodesic mapping; initial conditions; (pseudo-) Riemannian space
Summary:
Given a finite additive abelian group $G$ and an integer $k$, with $3\le k \le |G|$, denote by $\mathcal {D}_k (G)$ the simple incidence structure whose point-set is $G$ and whose blocks are the $k$-subsets $C = \lbrace c_1, c_2,\dots , c_k\rbrace $ of $G$ such that $c_1 + c_2+\dots +c_k = 0$. It is known (see [Caggegi, A., Di Bartolo, A., Falcone, G.: Boolean 2-designs and the embedding of a 2-design in a group arxiv 0806.3433v2, (2008), 1–8.]) that $\mathcal {D}_k (G)$ is a 2-design, if $G$ is an elementary abelian $p$-group with $p$ a prime divisor of $k$. From [Caggegi, A., Falcone, G., Pavone, M.: On the additivity of block design submitted.] we know that $\mathcal {D}_3(G)$ is a 2-design if and only if $G$ is an elementary abelian 3-group. It is also known (see [Caggegi, A.: Some additive $2-(v,4,\lambda )$ designs Boll. Mat. Pura e Appl. 2 (2009), 1–3.]) that $G$ is necessarily an elementary abelian 2-group, if $\mathcal {D}_4(G)$ is a 2-design. Here we shall prove that $\mathcal {D}_5(G)$ is a 2-design if and only if $G$ is an elementary abelian 5-group.
References:
[1] Beth, T., Jungnickel, D., Lenz, H.: Design Theory. 2nd ed., Cambridge University Press, Cambridge, 1999. MR 0890103 | Zbl 0945.05005
[2] Caggegi, A., Di Bartolo, A., Falcone, G.: Boolean 2-designs and the embedding of a 2-design in a group. arxiv 0806.3433v2, (2008), 1–8. MR 3468601
[3] Caggegi, A., Falcone, G., Pavone, M.: On the additivity of block design. submitted.
[4] Caggegi, A.: Some additive $2-(v,4,\lambda )$ designs. Boll. Mat. Pura e Appl. 2 (2009), 1–3. Zbl 1255.05028
[5] Colbourn, C. J., Dinitz, J. H.: The CRC Handbook of Combinatorial Designs. Discrete Mathematics and Its Applications, 2nd ed., Chapman & Hall/CRC Press, 2007. MR 2246267 | Zbl 1101.05001
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