Math 2.0

Group theory and generalizations Definition

Finitely generated group

Builds on Subgroup generated by a subset

Definition

A group GG is finitely generated when there is a finite subset SS of GG with ⟨S⟩=G\langle S \rangle = G, where ⟨S⟩\langle S \rangle is the subgroup generated by SS.

Remarks

Each finite group is finitely generated, because it is generated by the finite set of all its elements. The group (ℤn,+)(\mathbb{Z}^n, +) is finitely generated by the nn standard basis vectors. The group (ℚ,+)(\mathbb{Q}, +) is not finitely generated: the subgroup generated by finitely many fractions has a common denominator, and it does not contain all fractions.

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

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Group, concept, no evidence levelGroupconceptSubgroup, definition, no evidence levelSubgroupdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementSubgroup generated by a subset, definition, no evidence levelSubgroup generated by…definitionFinitely generated group, definition, no evidence levelFinitely generated gr…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Group concept
  2. Subgroup definition
  3. Subgroup criterion statement
  4. Intersection of subgroups is a subgroup statement
  5. Subgroup generated by a subset definition
  6. Finitely generated group definition

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Finitely_generated_group, lead section
  2. Web https://en.wikipedia.org/wiki/Finitely_generated_group, Examples
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