Minimal generating set

Definition
A generating set of a group is termed minimal or irredundant if any proper subset of the generating set, generates a strictly smaller (i.e. proper) subgroup. In other words, no generator can be dropped from the generating set.

Facts

 * Not every group may possess a minimal generating set. A group which possesses a minimal generating set is termed a minimally generated group
 * It is not in general necessary that two different minimal generating sets of a group, have the same cardinality. This is true for $$p$$-groups, viz., finite groups of prime power order. This follows from Burnside's basis theorem.
 * Any generating set of minimum cardinality (if finite) is a minimal generating set; the converse is not necessarily true.