I was reading the definition of a cyclic group $G$ which states:
A group $Q$ is cyclic if there is an element $a \in Q$ such that the subgroup generated by $a$ is the whole of $Q$. If $Q$ is a finite cyclic group with identity clement $e$, the set of elements of $Q$ may be written $\{e,a,a^2,\dots ,a^{n-1}\}$, where $a^n= e$ and $n$ is the smallest such positive integer.
I do not understand how $n$ can be the smallest such positive integer since $e$ is smaller than $a^{n-1}$. Maybe I misunderstood the definition. Could anyone explain?
Thanks!
Best Answer
The clue is in the name: it is cyclic. There is no order to the elements. But if you look at all powers of $a$, you will get $a^{i+kn}=a^i$ for all $k\in\Bbb Z$, $i\in\{0,\dots,n-1\}$, meaning that the powers are in a cycle. Note that $a^0=e$ by definition, but $0$ is not positive.