A property of the order of an element

#group_theory

Let $G$ be a group and $a \in G$ be an element. Let $n \in N$ be the smallest positive number such that $a^{n} = e$ , where $e$ is the identity element. Show that the set

{e, a, a^{2}, \dots, a^{n - 1}}

contains no repetitions.

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Let $G$ be a group and $a\in G$ be an element. Let $n\in {\bf N}$ be the smallest positive number such that $a^n=e$, where $e$ is the identity element. Show that the set
\[
	\{e,a,a^2,\ldots, a^{n-1}\}
\]
contains no repetitions.