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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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    http://math.stackexchange.com/questions/888211/notation-for-all-permutations-of-a-set. That link might be useful. I would just call it the set of permutations of a set. – 2017-02-09
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    This isn't quite right - the symmetric group has order $n!$, while the group I'm thinking of has order $n$. – 2017-02-09
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    I believe *cyclic permutation* applies here. I mean, I would usually refer to those as *cyclic permutations* of $(1, 2, \ldots, n)$. – 2017-02-09
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    That's also not quite right. Cyclic permutations are those which are periodic, this is something more specific. – 2017-02-09
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    If you're asking for the name of the *group*, I usually just see it referred to as "the cyclic group of order $n$." – 2017-02-09
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    Cyclic group of order $n$. You're just using a less common notation for its elements, just the second line of [Cauchy's two-line notation](https://en.wikipedia.org/wiki/Permutation#Definition_and_notations). You should check out [cycle notation](https://en.wikipedia.org/wiki/Permutation#Cycle_notation). – 2017-02-10

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