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 \({\mathbb{Z}_n}\).
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Presented by K. Kaarli.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Płonka, E. Weak automorphisms of dihedral groups. Algebra Univers. 64, 153–160 (2010). https://doi.org/10.1007/s00012-010-0096-x
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DOI: https://doi.org/10.1007/s00012-010-0096-x