Orthogonal and oriented Fano planes, triangular embeddings of $K_7,$ and geometrical representations of the Frobenius group $F_{21}$

Simone Costa, Marco Pavone
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Abstract

In this paper we present some geometrical representations of the Frobenius group of order $21$ (henceforth, $F_{21}$). The main focus is on investigating the group of common automorphisms of two orthogonal Fano planes and the automorphism group of a suitably oriented Fano plane. We show that both groups are isomorphic to $F_{21},$ independently of the choice of the two orthogonal Fano planes and of the choice of the orientation. We show, moreover, that any triangular embedding of the complete graph $K_7$ into a surface is isomorphic to the classical toroidal biembedding and hence is face $2$-colorable, with the two color classes defining a pair of orthogonal Fano planes. As a consequence, we show that, for any triangular embedding of $K_7$ into a surface, the group of the automorphisms that preserve the color classes is the Frobenius group of order $21.$ This way we provide three geometrical representations of $F_{21}$. Also, we apply the representation in terms of two orthogonal Fano planes to give an alternative proof that $F_{21}$ is the automorphism group of the Kirkman triple system of order $15$ that is usually denoted as #61.
正交和定向法诺平面、K_7$ 的三角嵌入以及弗罗贝纽斯群 $F_{21}$ 的几何表征
本文介绍了阶数为 $21$(以下简称 $F_{21}$)的弗罗贝纽斯群的一些几何表示。主要重点是研究两个正交法诺平面的共自变群和一个适当取向的法诺平面的自变群。我们证明这两个群都与 $F_{21}$ 同构,与两个正交法诺平面的选择和取向的选择无关。此外,我们还证明,任何将完整图 $K_7$ 嵌入曲面的三角形嵌入都与经典的环状双嵌入同构,因此是曲面 2$ 色的,两个色类定义了一对正交的法诺平面。因此,我们证明,对于任何将 $K_7$ 嵌入曲面的三角形,保持色类的自变量群是阶数为 $21 的弗罗贝尼斯群。此外,我们还利用两个正交法诺平面的表示,给出了$F_{21}$ 是阶为 $15$(通常表示为 #61)的柯克曼三元组的自变群的替代证明。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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