Let $ (G, *) $ be a group.
Say, $\langle a\rangle, \ \langle b\rangle$ are two cyclic subgroups of $G$ of equal orders.
Say $ \ o(a) = o(b) = n \in \mathbb N , \ n > 1 $
CASE I: $\{ a^p : p \in\mathbb N, \ \gcd(p,n) = 1\} \cap \{ b^p : p \in\mathbb N, \ \gcd(p,n) = 1\} = \phi$
CASE II: $\ \langle a\rangle \not\subset \langle b\rangle \text{ and }\langle b\rangle \not\subset \langle a\rangle$
In both cases I have an intution that $\langle a\rangle \cap \langle b\rangle \ = \ \{e\}$
Anyone, if you can please prove or counter it.