Math 2.0

Group theory and generalizations Theorem

Subgroups of a finite cyclic group correspond to divisors of its order

Builds on Cyclic groupSubgroups of cyclic groups are cyclicOrder of a power

Statement

Let GG be a cyclic group of finite order nn, generated by gg. For each positive divisor dd of nn, GG has exactly one subgroup of order dd, namely ⟨gn/d⟩\langle g^{n/d} \rangle. Each subgroup of GG has this form for some positive divisor dd of nn.

Remarks

Thus the number of subgroups of GG is the number of positive divisors of nn. For d,d′d, d' dividing nn, the subgroup of order dd is contained in the subgroup of order d′d' exactly when dd divides d′d'. For example, ℤ/12ℤ\mathbb{Z}/12\mathbb{Z} has six subgroups, of the orders 1,2,3,4,6,121, 2, 3, 4, 6, 12.

Reading path

To understand this page from the ground up, read these 10 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 levelSubgroupdefinitionOrder of an element, definition, no evidence levelOrder of an elementdefinitionCyclic subgroup generated by an element, definition, no evidence levelCyclic subgroup gener…definitiong to the n equals 1 iff the order divides n, statement, no evidence levelg to the n equals 1 i…statementCyclic group, definition, no evidence levelCyclic groupdefinitionOrder of a power, statement, no evidence levelOrder of a powerstatementSubgroups of cyclic groups are cyclic, statement, no evidence levelSubgroups of cyclic g…statementSubgroups of a finite cyclic group correspond to divisors of its order, statement, no evidence levelSubgroups of a finite…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. Order of an element definition
  5. Cyclic subgroup generated by an element definition
  6. g to the n equals 1 iff the order divides n statement
  7. Cyclic group definition
  8. Order of a power statement
  9. Subgroups of cyclic groups are cyclic statement
  10. Subgroups of a finite cyclic group correspond to divisors of its order statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cyclic_group, Subgroups
About this page
Tag0423TXR9AN
URIm2://0423TXR9AN
Aliasesfundamental-theorem-of-cyclic-groups
Kindstatement, theorem
WritingW0 stub
MSC20K01
StatusThis copy of the site has no build data. Run m2 status 0423TXR9AN.
UsesCyclic group, Subgroups of cyclic groups are cyclic, Order of a power

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