From Dummit & Foote, as usual, $\S$ 2.4 #14.
A group $H$ is called finitely generated if there is a finite set $A$ such that $H = \left \langle A \right \rangle$
(a) Prove that every finite group is finitely generated.
(b) (Prove that $\mathbb{Z}$ is finitely generated - I am comfortable with my proof of this, viz $\left \langle 1, -1 \right \rangle = \mathbb{Z}$ )
(c) Prove that every finitely generated subgroup of the additive group $\mathbb{Q}$ is cyclic [If $H$ is a finitely genereated subgroup of $\mathbb{Q}$, show that $H \leq \left \langle \frac{1}{k} \right \rangle$, where $k$ is the product of all the denominators which appear in a set of generators for $H$
Logically, I'm having a hard time getting started on both (a) and (c). For (a), I knew that $H$ generates itself (if that's the correct way to say it), i.e. $\left \langle G \right \rangle = G$ before looking at this relevant wikipedia page, but can't seem to articulate this in a manner consistent with the (currently inaccessible) definition of
$\left \langle A \right \rangle = \bigcap_{A\subseteq H; H \leq G} H$
or the proven result that this subgroup is the set of all products (... a bunch of elements in $G$ with exponents... "words", they call them)
Does this just follow simply from the above definition?
(c) I know (think?) that to show $H$ cyclic, I must take $h \in H$ and show that $h = a^k$, for some $k \leq |H|$ ... or something like that. But then there's that hint. Where do I begin?
Thanks for your help.