A group G is called cyclic if there is an element $a\in G$ such that the cyclic
subgroup generated by $a$ is the entire group $G$.Then
Now my confusion is that which one is correct ?
$G = \{a^n: \text{for all }n \in \mathbb{ Z}\}$.
$G = \{a^n: \text{for some }n \in \mathbb{ Z}\}$.
My thinking :If $G = \{a^n: \text{for some }n \in \mathbb{ Z}\}$,then $G$ will be a subgroup of $G$ this appears to contradict the definition of cyclic group .So $G = \{a^n: \text{for some }n \in \mathbb{ Z}\}$. is not correct
I think $G = \{a^n: \text{for all }n \in \mathbb{ Z}\}$ is correct
Best Answer
Actually, none of the options make sense. It should be$$G=\{a^n\mid n\in\Bbb Z\};$$which means that $G=\{\ldots,a^{-2},a^{-1},e_G,a,a^2,\ldots\}$.