Is there a name for the set $E(x, n)=\{x^p \mid p\in \mathbb N \land \space 0\leq p \leq n-1\}$ with $x\in G $, $G$ a group?

Source : Reversat & Bigonnet , Algèbre pour la licence ( = Undergraduate abstract algebra).

• The authors of the above mentionned book define the order of element $x\in G$ as the cardinal of $\langle x\rangle$ (i.e. $\langle \{x\}\rangle $) , that is, the cardinal of the subgroup generated by the set $\{x\} \subseteq G$ .

Note: So $\langle x\rangle$ is (by definition) the smallest subgroup of $G$ that contains $\{x\}$.

In symbols: $$ O(x)= Card (\langle x\rangle).$$

• Then they proceed to show that this definition of $O(x)$ is equivalent to the ( maybe more common) one:

$$ O(x)= n \iff ( x^n = e \land n= \min \{p\in \mathbb N^{*} \mid x^p = e\}) $$

with $e$ as the identity element of $G$.

• In order to prove this equivalence, they first define the set $E(x, m)$ as follows:

$$ E(x, n)=\{x^p \space \mid p\in \mathbb N \land \space 0\leq p \leq n-1\}.$$

Example: $E(x,4)= \{x^0, x^1, x^2, x^3 \}$ .

What I'd like to know is whether there is a common name for $E(x, n)$, and whether there is a symbol communly used to denote this set.

A common name for $E(x,n)$, considered as a group, is $C_n$, the cyclic group of order $n$. So $$ C_n=\{e,x, x^2,\ldots ,x^{n-1}\}, $$ where $x^n=x^0=e$.

In the context of your question, $C_n\cong \langle x\rangle$, where $x$ has order $n$, e.g., $x^n=e$, but $x^m\neq e$ for $1\le m<n$.

It is also common to write this by the presentation $$ C_n=\langle x\mid x^n=e\rangle. $$ The "C" stands for "cyclic". Another example of a group presentation is, for the dihedral group $D_n$, $$ D_n=\{r,s\mid r^n=s^2=e, srs=r^{-1}\rangle, $$ for $n\ge 3$. This is not a cyclic group in general.