On commutation semigroups of dihedral groups. (Q382941)

Let \(G\) be a group. For every \(g\in G\) two kinds of transformations of \(G\) are considered: \(\rho_g(x)=[x,g]\), \(\lambda_g(x)=[g,x]\). Let \(\mathbf P(G)\) and \(\Lambda(G)\) be the semigroups of transformations generated by the sets \(\{\rho_g:g\in G\}\) and \(\{\lambda_g:g\in G\}\), respectively. Explicit formulas for the orders of \(\mathbf P(G)\) and \(\Lambda(G)\) are developed when \(G=D_m\), the dihedral group of order \(2m\). These formulas are applied to determining when the orders of \(\mathbf P(G)\) and \(\Lambda(G)\) coincide or these semigroups are isomorphic.