Bureaucrats, emailconfirmed, Administrators

38,774

edits
Jump to: navigation, search

→Symbol-free definition

A [[subset of a group]] is termed a '''generating set''' if it satisfies the following equivalent conditions:

* Every element of the group can be expressed as a [[defining ingredient::word]] in terms of the elements of this subset, i.e., it can be expressed using the elements of the subset by means of the group operations of multiplication and inversion. (note that if the subset is a [[symmetric subset]], i.e., it contains the identity and is closed under taking inverses, then every element of the group must be a product of elements in the subset. Symmetric subsets arise, for instance, when we take a union of subgroups).

* There is no proper subgroup of the group containing this subset<section end=beginner/>

* There is a surjective map from a [[free group]] on that many generators to the given group, that sends the generators of the free group to the elements of this ''generating set''.

Retrieved from "https://groupprops.subwiki.org/wiki/Special:MobileDiff/42040"