Group theory and generalizations Foundations Concept

Generators of a group

Builds on Subgroup generated by a subset

Summary

A subset $S$ of a group $G$ generates $G$ when the subgroup generated by $S$ is all of $G$ . Then each element of $G$ is a finite product of elements of $S$ and of their inverses. The elements of $S$ are called generators of $G$ .

Remarks

The set $\{1\}$ generates $(\mathbb{Z}, +)$ , and so does the set $\{2, 3\}$ . Each group is generated by the set of all its elements. A group generated by one element is a cyclic group, and a group generated by a finite set is a finitely generated group.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Group concept
Subgroup definition
Subgroup criterion statement
Intersection of subgroups is a subgroup statement
Subgroup generated by a subset definition
Generators of a group concept

Tag	`0MBV2VHMXA`
URI	`m2://0MBV2VHMXA`
Aliases	`generating-set-of-a-group`
Kind	concept
Writing	W1 stated
MSC	20F05
Uses	Subgroup generated by a subset
Same as	Wikidata Q734209 Wikipedia Generating set of a group

Generators of a group

Summary

Remarks

Reading path

Further reading

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