When do different cyclic subgroups of same order share the neutral element only?

When do different cyclic subgroups of same order share the neutral element only?

They do if their order is prime as each element of each group (but the neutral one) generates the hole group; therefore, if distinct elements exist the groups must be different.

Is there a more general criterion?

• Same thing applies. A cyclic subgroup is still a group itself. Any two cyclic (sub)groups of order $n$ are unique. Technically I should say unique up to isomorphism. So if you have two cyclic (sub)groups that don't look identical, they are still isomorphic. – tilper Mar 31 '17 at 13:29
• @tilper Different doesn't mean non-isomorphic, e.g. if $G=\mathbb{Z}_2\oplus\mathbb{Z}_2$ then $G$ has 3 different cyclic subgroups of order $2$, namely $\mathbb{Z}_2\oplus 0$, $0\oplus\mathbb{Z}_2$ and the diagonal $\{(0,0), (1,1)\}$. He's talking about counting elements of two subgroup of some group obviously. – freakish Mar 31 '17 at 13:36