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We show that symmetric block designs \({\mathcal {D}}=({\mathcal {P}},{\mathcal {B}})\) can be embedded in a suitable commutative group \({\mathfrak {G}}_{\mathcal {D}}\) in such a way that the sum of the elements in each block is zero, whereas the only Steiner triple systems with this property are the point-line designs of \({\mathrm {PG}}(d,2)\) and \({\mathrm {AG}}(d,3)\). In both cases, the blocks can be characterized as the only k-subsets of \(\mathcal {P}\) whose elements sum to zero. It follows that the group of automorphisms of any such design \(\mathcal {D}\) is the group of automorphisms of \({\mathfrak {G}}_\mathcal {D}\) that leave \(\mathcal {P}\) invariant. In some special cases, the group \({\mathfrak {G}}_\mathcal {D}\) can be determined uniquely by the parameters of \(\mathcal {D}\). For instance, if \(\mathcal {D}\) is a 2-\((v,k,\lambda )\) symmetric design of prime order p not dividing k, then \({\mathfrak {G}}_\mathcal {D}\) is (essentially) isomorphic to \(({\mathbb {Z}}/p{\mathbb {Z}})^{\frac{v-1}{2}}\), and the embedding of the design in the group can be described explicitly. Moreover, in this case, the blocks of \(\mathcal {B}\) can be characterized also as the v intersections of \(\mathcal {P}\) with v suitable hyperplanes of \(({\mathbb {Z}}/p{\mathbb {Z}})^{\frac{v-1}{2}}\). 相似文献
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Andrea Caggegi 《Geometriae Dedicata》1985,18(3):239-247
This paper presents an example of an infinite generalized André plane with the following properties: (1) has dimension eight over its kernel (hence is neither desarguesian nor a Hall plane); (2) the full collineation group of has no orbit of finite length on the line at infinity.
Lavoro eseguito nell'ambito delle attività del G.N.S.A.G.A. del C.N.R. 相似文献
Lavoro eseguito nell'ambito delle attività del G.N.S.A.G.A. del C.N.R. 相似文献
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Andrea Caggegi 《Annali di Matematica Pura ed Applicata》1981,128(1):169-191
Summary In this paper I classify primitive Hestenes ternary rings having minimal onesided ideals.
L'autore appartiene al G.N.S.A.G.A. del C.N.R. 相似文献
L'autore appartiene al G.N.S.A.G.A. del C.N.R. 相似文献
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The simple incidence structure , formed by the points and the unordered pairs of distinct parallel lines of a finite affine plane of order n > 4, is a 2 – (n
2,2n,2n–1) design with intersection numbers 0,4,n. In this paper, we show that the converse is true, when n ≥ 5 is an odd integer.
Supported by M.I.U.R., Università di Palermo. 相似文献
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