# Minimal generating set

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## Contents

## 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 -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.