Minimal generating set

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This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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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. Further information: Group with fixed size of minimal generating set
  • Any generating set of minimum cardinality (if finite) is a minimal generating set; the converse is not necessarily true.