Weak automorphisms of dihedral groups. (Q616133)

Let \(G\) be a group; a bijective map \(\tau\colon G\to G\) is called a weak automorphism of \(G\) if for every \(n\in\mathbb N\), every \(g_1,g_2,\dots,g_n\in G\) and every \(\alpha_1,\alpha_2,\dots,\alpha_n\in\mathbb Z\) there are \(\beta_1,\beta_2,\dots,\beta_n\in\mathbb Z\) such that \[ \tau(g_1^{\alpha_1}\cdot g_2^{\alpha_2}\cdots g_n^{\alpha_n})=\tau(g_1)^{\beta_1}\cdot\tau(g_2)^{\beta_2}\cdots\tau(g_n)^{\beta_n}. \] In this paper the author describes in full details the structure of \(\mathrm{WAut}(D_n)\), the weak automorphism group of a dihedral group \(D_n\) of order \(2n\).