A group with a finitely generated subgroup of finite index

Question

$G$ is a group, and $H$ is a subgroup of $G$ with index $[G:H]=n$. Prove or disprove the following:

1. If $H$ is a finitely generated group then $G$ is a finitely generated group.
2. If $a\in G$ then $a^n\in H$.
3. If $a \in G$ then $H\cap \{a,a^2,...,a^n\}\ne \emptyset$.

Progress

1. If $G$ is finite, then $G$ is a finitely generated group (as all finite groups). If $G$ isn't finite, then due to the finite index, I conclude that $H=G$, and then $G$ is a finitely generated group.
2. Thank you @lhf for the link.
3. I need help here.

Answer 1

(1) Let $H=\langle K\rangle, |K|<\infty$ and $\{x_1,\ldots,x_n\}$ be representatives of the cosets of $H$. Then $G=\langle K\cup\{x_1,\ldots,x_n\}\rangle$ is finitely generated.

(3) Let $H\cap \{a,a^2,...,a^n\}= \emptyset$. Then all of $a,a^2,...,a^n$ are contained in different cosets, and every coset $a^iH$ is different from $H$. So there are $\ge n+1$ cosets, this contradicts $[G:H]=n$.