Is there a name for the set E(x,n)={xp∣p∈N∧≤p≤n−1} with x∈G,G a group?

Question

Is there a name for the set E(x,n)={xp∣p∈N∧≤p≤n−1} with x∈G,G a group?

regresavo552 · Accepted Answer

A common name for E(x,n), considered as a group, is

C_{n}

, the cyclic group of order n. So

C_{n} = {e, x, x^{2}, \dots, x^{n - 1}}

,
where

x^{n} = x^{0} = e

.
In the context of your question,

C_{n} \tilde{=} ⟨ x ⟩

, where x has order n, e.g.,

x^{n} = e, but x^{m} \neq e for 1 \leq m < n

.
It is also common to write this by the presentation

C_{n} = ⟨ x ∣ x^{n} = e ⟩

.
The "C" stands for "cyclic". Another example of a group presentation is, for the dihedral group

D_{n}

,

D_{n} = {r, s ∣ r^{n} = s^{2} = e, s r s = r^{- 1}}

,
for

n \geq 3

. This is not a cyclic group in general.

Is there a name for the set E(x,n)=\{x^{p}|p \in N