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I just wanted to know whether there is any standard topology on groups like $\mathbb{Z}/n\mathbb{Z}$ or $\mathbb{Z}$ ? - The only one that I could imagine, especially for finite groups is the discrete topology. Would you agree that the discrete topology is the standard one in the context of covering maps and so on?

EDIT: It was suggested in the comments that this is true, so I would also be interested in the question: Does the discrete topology always mean include that the group is a topological one? -I actually think that this is true.

  • Do you require that the topology turns the group into a topological group? – zarathustra Jul 27 '14 at 12:49
  • If there's no obvious topology, the group is usually given the discrete topology yes. – Najib Idrissi Jul 27 '14 at 13:04
  • Topological groups impose continuity of the group operation (and commonly on group actions if applicable). This previous Math.SE Question, Covering of a topological group is a topological group, may be of interest. – hardmath Jul 27 '14 at 13:07
  • What is the topology induced by the metric on the cayley graph out of interest? – JC574 Jul 27 '14 at 13:23
  • @JC574: Since the Cayley graph metric evaluated on pairs of vertices always has values in the non-negative integers, the radius $1/2$ open ball in the group around any element consists solely of that element. So the topology on the group is discrete. – Lee Mosher Jul 27 '14 at 13:28
  • cool, thanks. should have just thought about it really haha – JC574 Jul 27 '14 at 13:34
  • @FPE actually no, but aren't both groups topological groups with the discrete topology? –  Jul 27 '14 at 14:40
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    Why would you put a topology on a group that has nothing to do with the group structure? Anyway, a finite topological group is always discrete "mod $N$" where $N$ is the closure of the identity, a normal subgroup. See here. We can also regard $\Bbb Z/n\Bbb Z$ and $\Bbb Z$ as subgroups of $S^1$ and $\Bbb R$ respectively, and as subspaces they are discrete. Usually the "standard topology" is determined by context and established historical precedent. – anon Jul 27 '14 at 14:52
  • @blue thank you, i see. –  Jul 27 '14 at 14:55

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