Math 2.0

Group theory and generalizations Theorem

Subgroups of cyclic groups are cyclic

Builds on Cyclic groupSubgroup

Statement

Let GG be a cyclic group and let HH be a subgroup of GG. Then HH is cyclic.

Remarks

Let gg generate GG. If H≠{e}H \neq \{e\}, let mm be the smallest positive integer with gm∈Hg^m \in H; then division with remainder shows that H=⟨gm⟩H = \langle g^m \rangle. For example, each subgroup of (ℤ,+)(\mathbb{Z}, +) has the form mℤm\mathbb{Z}, generated by mm.

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, no evidence levelGroupconceptLaws of exponents in a group, statement, no evidence levelLaws of exponents in …statementSubgroup, definition, no evidence levelSubgroupdefinitionCyclic subgroup generated by an element, definition, no evidence levelCyclic subgroup gener…definitionCyclic group, definition, no evidence levelCyclic groupdefinitionSubgroups of cyclic groups are cyclic, statement, no evidence levelSubgroups of cyclic g…statement
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.
  • no evidence level
  1. Group concept
  2. Laws of exponents in a group statement
  3. Subgroup definition
  4. Cyclic subgroup generated by an element definition
  5. Cyclic group definition
  6. Subgroups of cyclic groups are cyclic statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cyclic_group, Subgroups
About this page
Tag0KB1ND7XDS
URIm2://0KB1ND7XDS
Kindstatement, theorem
WritingW0 stub
MSC20K27
StatusThis copy of the site has no build data. Run m2 status 0KB1ND7XDS.
UsesCyclic group, Subgroup

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