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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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.