# 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

## 1 Answer

Too long for a comment:

1) I don't think it is important (under what considerations?), but it makes things simpler in many cases.

For (2) you may want to check more closely Burnside's Problem

(3) Yes, but sometimes there may be other definitions for rank. One has to check carefully what does a specific author means and within what context.