→Types of generating sets
| finite generating set || The generating set is finite as a set || [[finitely generated group]]
| [[minimal generating set]] || The generating set has no proper subset that is also a generating set || [[minimally generated group]]
| generating set of minimum size || A generating set such that there is no generating set of smaller size. If finite, then this must also be a minimal generating set ||
It is not in general true that any two minimal generating sets of a group have the same size. However, two important related facts are true:
* If a group has a finite generating set, then every generating set has a finite subset that is a generating set, and in particular, every minimal generating set is finite. For more, see [[equivalence of definitions of finitely generated group]].
* We can consider the property of being a [[group in which all minimal generating sets have the same size]]. Any [[group of prime power order]] has this property. This follows from [[Burnside's basis theorem]].
==Study of this notion==