Bureaucrats, emailconfirmed, Administrators

38,822

edits
Jump to: navigation, search

no edit summary

The generators of a group may not in general be independent. That is, there may be words in the generators that simplify to the identity in the given group. Such a word is termed a [[relation]] between the generators. Relations can be viewed as elements of the free group on symbols corresponding to the generators. Thus relations form a [[normal subgroup]] of the free group. Specifying a set of relations whose normal closure is this normal subgroup is what we call a [[presentation of a group]].

==Types of generating sets==

{| class="sortable" border="1"

! Type of generating set !! Meaning !! Property of being a group that has such a generating set

|-

| finite generating set || The generating set is finite as a set || [[finitely generated group]]

|-

| [[minimal 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. 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]].

==Study of this notion==

Retrieved from "https://groupprops.subwiki.org/wiki/Special:MobileDiff/50210"