If a generated subgroup is cyclic

Question

I would like to make a similar question to question "Exercise on generated subgroup":

Let $G$ be a finite group and $H\leq G$, $H$ cyclic with $|H|=exp(G)$. If $x\in C_{G}(H)\smallsetminus H$, then is $\langle x,H\rangle $ cyclic? If this statement is true, how to prove it?

Answer 1

No - consider $G = C_2 \times C_2$, and $H$ an index 2 subgroup.

Answer 2

You don't really need to provide a counterexample. It is not possible for $\langle x, H \rangle$ to be cyclic given the other conditions, even without $x \in C_{G}(H)$. Since $|H| = \exp(G),$ the only way that $\langle x,H \rangle$ could be cyclic would be if $\langle x,H \rangle = H,$ which would force $x \in H,$ but you have assumed $x \not \in H.$