Steiner loops of affine type are associated to arbitrary Steiner triple systems. They behave to elementary abelian 3-groups as arbitrary Steiner Triple Systems behave to affine geometries over $${\mathrm {GF}}(3)$$ . We investigate algebraic and geometric properties of these loops often in connection to configurations. Steiner loops of affine type, as extensions of normal subloops by factor loops, are studied. We prove that the multiplication group of every Steiner loop of affine type with n elements is contained in the alternating group $$A_n$$ and we give conditions for those loops having $$A_n$$ as their multiplication groups (and hence for the loops being simple).

中文翻译：

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仿射型 Steiner 环

仿射类型的 Steiner 环与任意 Steiner 三元系统相关联。它们对基本阿贝尔 3 群表现得像任意 Steiner 三重系统对 $${\mathrm {GF}}(3)$$ 上的仿射几何表现。我们研究这些环的代数和几何特性，通常与配置有关。研究了仿射类型的 Steiner 环，作为正常子环的因子环的扩展。我们证明了每个具有 n 个元素的仿射类型的 Steiner 循环的乘法组都包含在交替组 $$A_n$$ 中，并且我们为那些以 $$A_n$$ 作为乘法组的循环给出了条件（因此对于循环很简单）。