Rank of group-3

For any group $G$ and subset $S\subseteq G$, $\langle S\rangle$ denotes the subgroup generated by $S$ and rank of G, $d(G):=\min\{\lvert S\rvert ;S\subseteq G,\langle S\rangle =G\}$

My questions about rank are: (consider $G$ infinite if necesssary)

1) Why is it important group have finite rank?

2) There is some relationship between the rank and the exponent of a group G ?

3) Is the above definition general? Is the same for abelian groups, free, abelian free, nilpotent?

• Re (3): This is a definition, so it is applicable to any group. That is might be completely useless in some cases doesn't matter (no, I'm not saying this definition is useless in any case). – vonbrand Mar 10 '13 at 13:20
• Re 1 and 3: The classification of abelien groups of finite rank is quite easy (finite direct sum of cyclic groups), but not quite so for infinite rank and not so for nonabelian groups. – Hagen von Eitzen Mar 10 '13 at 13:26