Structure of a group related to a Napoleon tessellation (Q372502)

In this paper, the author presents a comprehensive description of the structure of the polyhedral group (G) using a geometric representation. He demonstrates two interesting theorems: 1) Let \(S_{N}\) be the symmetry group of the Napoleon tesselation. The mapping \( {\Psi}:G{\rightarrow} S_{N},{\Psi}(a)=R_{(0,0)} \) and \({\Psi}(b)=R_{(0,1)}\) is an isomorphism of the groups G and \(S_{N}\); 2) If H is the free abelian group of rank 2 and K is the cyclic group of order 3, then H corresponds to the normal subgroup of all translations of \(S_{N} \) and \(K=<{\Psi}^{-1}(R_{(0,0)})>\).