Weak automorphisms of dihedral groups

Abstract

Let \({\mathcal{A} = (A; F)}\) be an algebra with T the set of all its term operations. For any permutation τ of A, the induced mapping \({f \to \tau\circ f\circ\tau^{-1}}\) defines a permutation \({\tau^{\star}}\) of the set of all finitary operations on the set A. We say that τ is a weak automorphism of \({\mathcal{A}}\) if and only if τ*(T) = T. Of course any automorphism α of \({\mathcal{A}}\) is a weak automorphism, because α*(t) = t for all \({t \in T}\). The set of all weak automorphisms of \({\mathcal{A}}\) forms a subgroup of the symmetric group on A. In this paper, we describe weak automorphisms of the dihedral groups \({\mathcal{D}_n}\) for n ≥ 3. We show that the weak automorphism group of \({\mathcal{D}_n}\) is a semidirect product of the group of automorphisms of \({\mathcal{D}_n}\) and some group related to the group of invertible elements of the ring \({\mathbb{Z}_n}\).