# 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?

• Different cyclic groups of the same order? Cyclic groups of a fixed order are unique. – tilper Mar 31 '17 at 13:26
• Sorry, I am speaking of cyclic subgroups of a group. – Ramen Mar 31 '17 at 13:28
• 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
• I am not concerned with isomorphism. I want to count elements. – Ramen Mar 31 '17 at 13:32
• @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