Consider the set of positive integers as a group with its binary operation being addition. Is this group's set of symmetries countable or uncountable?


closed as unclear what you're asking by Mike Earnest, Connor Harris, Dietrich Burde, Derek Holt, José Carlos Santos May 22 '18 at 17:34

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  • $\begingroup$ Do you mean its automorphism group? That's finite. $\endgroup$ – Lord Shark the Unknown May 22 '18 at 15:31
  • $\begingroup$ For symmetric group see this question. For symmetry group of the integers see here. $\endgroup$ – Dietrich Burde May 22 '18 at 15:33
  • $\begingroup$ Dear Mr. Burde, I was actually thinking of the set of positive integers as a dihedral group rather than a symmetric group. $\endgroup$ – Lorenzo Gil Badiola May 22 '18 at 15:38
  • $\begingroup$ The positive integers are a semigroup, not a group. There are no inverses. $\endgroup$ – Robert Israel May 22 '18 at 15:40
  • $\begingroup$ Since the number of symmetries of dihedral group N is twice the number of its elements, shouldn't the number of symmetries of this infinite dihedral group be double that of the set of natural numbers? $\endgroup$ – Lorenzo Gil Badiola May 22 '18 at 15:41

You may be thinking of a cycle graph, either finite or infinite. The symmetries of this graph is the dihedral group with twice the number of vertices of the cycle graph. Note that this is not the positive integers, which are not a group, since you don't have a graph structure compatible with the adjacency of the positive integers since $1$ is only adjacent to $2$.

However, you can consider all the integers which is a group, as an infinite cycle graph and its symmetry group is the infinite dihedral group.


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