cyclic group and its generators 
In a cyclic group $G$, let $g1$, $g2$ be two distinct generators. Then $g1g2$ is also a generator of $G$.

Does the above statement hold true? It does not mention in the question that $G$ is finite or infinite.
please help.
 A: This is not true in general. In ${\mathbb Z_n}$, both $1$ and $n-1$ are generators, but the sum of these two elements is the identity, which is not a generator.
In the infinite case, if $G=\langle g \rangle$ is an infinite cyclic group, then $g$ and $g^{-1}$ are the only two generators and once again their product is the identity, not a generator.
A: In $\mathbb{Z}_n$ or in $\mathbb{Z}$, $1$ and $-1$ are generators whereas $1-1=0$ is not.
A: Both Seirios' and Zach's answers illustrate a more general point. Indeed if $G$ is a group such that $G=\langle g\rangle$ (so $G$ is generated by $g$), then $G$ will also be generated by the element $g^{-1}$, and so assuming that $g\neq g^{-1}$ (which occurs precisely when $G$ is the cyclic - and unique - group of order 2), then this gives you a counterexample. 
If the group that you are dealing with is cyclic of order $n$, then if $G=\langle g_{1}\rangle$ and $g_{2}=g_{1}^{k}$ for some $k$, then $g_{1}g_{2}$ will be a generator for $G$ if and only if $k+1$ is coprime to $n$.
For infinite cyclic groups, see Zach's answer.
