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Let $C_n$ be the group generated by the permutation $(1, 2, ..., n-1, n)$:

$$ C_1 = \{ (1) \} \\C_2 = \{ (1, 2), (2, 1) \} \\C_3 = \{ (1, 2, 3), (2, 3, 1), (3, 1, 2) \} \\C_4 = \{ (1, 2, 3, 4), (2, 3, 4, 1), (3, 4, 1, 2), (4, 1, 2, 3) \} \\ ... $$

Is there a commonly accepted name for $C_n$?

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  • $\begingroup$ math.stackexchange.com/questions/888211/…. That link might be useful. I would just call it the set of permutations of a set. $\endgroup$ – lordoftheshadows Feb 9 '17 at 19:21
  • $\begingroup$ This isn't quite right - the symmetric group has order $n!$, while the group I'm thinking of has order $n$. $\endgroup$ – Carl Patenaude Poulin Feb 9 '17 at 19:24
  • $\begingroup$ I believe cyclic permutation applies here. I mean, I would usually refer to those as cyclic permutations of $(1, 2, \ldots, n)$. $\endgroup$ – pjs36 Feb 9 '17 at 19:30
  • $\begingroup$ That's also not quite right. Cyclic permutations are those which are periodic, this is something more specific. $\endgroup$ – Carl Patenaude Poulin Feb 9 '17 at 19:32
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    $\begingroup$ If you're asking for the name of the group, I usually just see it referred to as "the cyclic group of order $n$." $\endgroup$ – pjs36 Feb 9 '17 at 19:38

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