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

Simone Costa; Marco Pavone; Simone Costa; Marco Pavone

doi:10.3934/math.20241676

AIMS Mathematics

2024, Volume 9, Issue 12: 35274-35292. doi: 10.3934/math.20241676

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

Simone Costa ¹,
Marco Pavone ^{2
,
,}

1.
DICATAM - Sez. Matematica, Università degli Studi di Brescia, Brescia 25123, Italy
2.
Dipartimento di Ingegneria, Università degli Studi di Palermo, Palermo 90128, Italy

Received: 06 July 2024 Revised: 23 November 2024 Accepted: 03 December 2024 Published: 18 December 2024
MSC : 05C25, 05B07, 05C76, 05C10

In this paper we present some geometrical representations of $ F_{21}, $ the Frobenius group of order $ 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 orientation.
Moreover, since any triangular embedding of the complete graph $ K_7 $ into a surface is isomorphic, as is well known, to the classical (face $ 2 $-colorable) toroidal biembedding, and since the two color classes define a pair of orthogonal Fano planes, we deduce, as an application of our previous result, that the group of the embedding automorphisms that preserve the color classes is the Frobenius group of order $ 21. $
In this way, we provide three geometrical representations of $ F_{21} $. Also, we apply once more 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, thereby confirming again the potential of our Fano-plane approach.
Although some of the results in this paper may be (partially) known, we include direct and independent proofs in order to make the paper self-contained and offer a unified view on the subject.
- orthogonal Fano planes,
- oriented Fano plane,
- Frobenius group $ F_{21} $,
- toroidal embedding,
- Kirkman triple system,
- STS($ 15 $), KTS($ 15 $)
Citation: Simone Costa, Marco Pavone. Orthogonal and oriented Fano planes, triangular embeddings of $ K_7, $ and geometrical representations of the Frobenius group $ F_{21} $[J]. AIMS Mathematics, 2024, 9(12): 35274-35292. doi: 10.3934/math.20241676

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Abstract

In this paper we present some geometrical representations of $ F_{21}, $ the Frobenius group of order $ 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 orientation.

Moreover, since any triangular embedding of the complete graph $ K_7 $ into a surface is isomorphic, as is well known, to the classical (face $ 2 $-colorable) toroidal biembedding, and since the two color classes define a pair of orthogonal Fano planes, we deduce, as an application of our previous result, that the group of the embedding automorphisms that preserve the color classes is the Frobenius group of order $ 21. $

In this way, we provide three geometrical representations of $ F_{21} $. Also, we apply once more 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, thereby confirming again the potential of our Fano-plane approach.

Although some of the results in this paper may be (partially) known, we include direct and independent proofs in order to make the paper self-contained and offer a unified view on the subject.

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