Minimal generating sets for a group may have different sizes
- is a Minimal generating set (?) for , i.e., is a generating set for and removing any element from gives something that is not a generating set for .
- is also a minimal generating set for .
- The cardinalities of and are not equal.
A group where all minimal generating sets have the same size is termed a group in which all minimal generating sets have the same size. By Burnside's basis theorem, it turns out that any group of prime power order satisfies the property. There do exist groups that are not of prime power order which also satisfy the property; for instance, the symmetric group:S3 and more generally any dihedral group of odd prime degree has the property that every minimal generating set has size two.
Example of finite cyclic group of order six
Let be the group of integers modulo 6 under addition. Consider and . Both and are minimal generating sets and they have different sizes.