[Math] Is it possible for a cyclic additive group to have more than one generator

Question

I'm trying to find an element $k$ that generates the cyclic additive group $\mathbb{Z}_{6}$. Since a group is cyclic, the entire group can be generated by a single element. I've tried adding $\langle1\rangle$ and $\langle5\rangle$ repeatedly in modulo $6$. And both $1$ and $5$ give me all the elements of $\mathbb{Z}_{6}$. So is it possible for a cyclic additive group to have more than one generator or am I doing something wrong?

Answer 1

You're absolutely right. More generally, if you have an element $a$ that generates a finite cyclic group $G$, the group is also generated by another element $a^n$ (in multiplicative notation, $n\cdot a$ in additive notation) iff $n$ is coprime to $|G|$, i.e. $\gcd(n,|G|)=1$. This is because in that case, you can write any exponent $k$ in $a^k$ as $k=rn+s|G|$, and thus

$$a^k=a^{rn+s|G|}=\left(a^n\right)^r+\left(a^s\right)^{|G|}=\left(a^n\right)^r\;.$$