Let $G$ be a cyclic group of order $n$. If $a$ and $a^2$ are both generators of $G$. Then what can we say about n?
There are three options:
- $3$ must be co-prime to $n$.
- $n$ is divisible by $3$.
- $3$ and $n$ may not prime integers.
I can show that the statement $2$ is incorrect. But which is the correct answer then? Please help me to solve this.
In general, $a^2$ will generate a subgroup of index $2$ if $n$ is even and the whole group if $n$ is odd. Thus the correct answer is
4) $n$ is odd.