A question in Isomorphism

Question

Let G be a cyclic group. Soppose G and G' are isomorphic groups. Show that G' is also cyclic.

Can Someone Solve this pleaase? I have an exam 2 hours later!

Answer 1

If a group $G$ of order $n$ is cyclic, then that is equivalent to saying that there exists an element $g \in G$ such that $g^n = e$, where $n$ is the smallest natural number such that this holds.

(Note that $e$ is the identity element)

Answer 2

Put differently, a group $G$ is cyclic if it can be generated by a single one of its elements. If $G$ is our group, and $g$ is a generator element of $G$, then for any other $g' \in G$ we have that $g = (g')^k$ for some $k \in \mathbb{Z}$. We can also write this as $G = \left\langle g \right\rangle$.