@article{Costa2024, 
author = {Simone Costa and Marco Pavone},
title = {Orthogonal and oriented Fano planes, triangular embeddings of  K7, and geometrical representations of the Frobenius group  F21},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {12},
pages = {35274-35292},
keywords = {orthogonal Fano planes, oriented Fano plane, Frobenius group F21, toroidal embedding, Kirkman triple system, STS(15), KTS(15)},
url = {https://www.sciopen.com/article/10.3934/math.20241676},
doi = {10.3934/math.20241676},
abstract = {In this paper we present some geometrical representations of  F21, 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  F21, independently of the choice of the two orthogonal Fano planes and of the orientation.Moreover, since any triangular embedding of the complete graph  K7 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  F21. Also, we apply once more the representation in terms of two orthogonal Fano planes to give an alternative proof that  F21 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.}
}