# Set of rotations of an ordered tuple

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 $C_n$?

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