Changes

Jump to: navigation, search

Generating set of a group

509 bytes added, 04:47, 11 April 2017
Types of generating sets
| finite generating set || The generating set is finite as a set || [[finitely generated group]]
|-
| [[minimal generating set]] (also known as irredundant 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(see [[. 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]].
 
== Associated arithmetic functions ==
 
* [[Minimum size of generating set]]: This is the smallest possible cardinality of a generating set for the group. It is finite if and only if the group is a [[finitely generated group]].
* [[Maximum size of minimal generating set]]: This is the largest possible cardinality of a minimal generating set for the group. It is defined for a [[fintie group]], but not necessarily for an infinite finitely generated group.
==Study of this notion==
Bureaucrats, emailconfirmed, Administrators
38,756
edits

Navigation menu