Intrinsic Geometries of the Simple Group of Order 168 (Q6653380)

Among the finite groups of order \(168\), there is, up to isomorphism, a unique simple group \(G\). Interestingly, it has a twofold nature, as \(\mathrm{PSL}(2,7) \simeq G \simeq \mathrm{PSL}(3,2)\). In this paper the author show that \(G\) has a natural action on the projective line over \(\mathbb{F}_{7}\), so it is isomorphic to \(\mathrm{PSL}(2,7)\), and a natural action on the projective plane over \(\mathbb{F}_{2}\), so it is isomorphic to \(\mathrm{PSL}(3,2)\). The proof relies solely on elementary methods of group theory.