Math 2.0

Group theory and generalizations Proposition

A finite abelian group is cyclic iff its exponent equals its order

Builds on Exponent of a groupCyclic groupAbelian groupClassification of finite abelian groupsOrder of an element

Statement

Let GG be a finite abelian group. Then GG is cyclic if and only if the exponent of GG equals ∣G∣|G|.

Remarks

If GG is cyclic with generator gg, then the order of gg is ∣G∣|G|, so the exponent is ∣G∣|G|. For the converse, write G≅ℤ/k1ℤ⊕⋯⊕ℤ/kuℤG \cong \mathbb{Z}/k_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/k_u\mathbb{Z} with 1<k1∣k2∣⋯∣ku1 < k_1 \mid k_2 \mid \cdots \mid k_u, by the classification. The exponent is kuk_u and the order is k1⋯kuk_1 \cdots k_u, so they are equal only when u≤1u \leq 1. The abelian condition is necessary: S3S_3 has exponent 66 and order 66, but it is not cyclic. For example, (ℤ/nℤ)×(\mathbb{Z}/n\mathbb{Z})^\times is cyclic if and only if λ(n)=φ(n)\lambda(n) = \varphi(n).

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

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Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptCommutativity, definition, no evidence levelCommutativitydefinitionDirect product of groups, definition, no evidence levelDirect product of gro…definitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionLaws of exponents in a group, statement, no evidence levelLaws of exponents in …statementOrder of a group, definition, no evidence levelOrder of a groupdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionGroup isomorphism, definition, no evidence levelGroup isomorphismdefinitionOrder of an element, definition, no evidence levelOrder of an elementdefinitionCyclic subgroup generated by an element, definition, no evidence levelCyclic subgroup gener…definitionExponent of a group, definition, no evidence levelExponent of a groupdefinitiong 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 cyclic subgroup equals the order of its generator, statement, no evidence levelOrder of a cyclic sub…statementCyclic groups of the same order are isomorphic, statement, no evidence levelCyclic groups of the …statementProduct of cyclic groups is cyclic iff the orders are coprime, statement, no evidence levelProduct of cyclic gro…statementClassification of finite abelian groups, statement, no evidence levelClassification of fin…statementA finite abelian group is cyclic iff its exponent equals its order, statement, no evidence levelA finite abelian grou…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. Binary operation definition
  2. Group concept
  3. Commutativity definition
  4. Direct product of groups definition
  5. Group homomorphism definition
  6. Laws of exponents in a group statement
  7. Order of a group definition
  8. Subgroup definition
  9. Abelian group definition
  10. Group isomorphism definition
  11. Order of an element definition
  12. Cyclic subgroup generated by an element definition
  13. Exponent of a group definition
  14. g to the n equals 1 iff the order divides n statement
  15. Cyclic group definition
  16. Order of a cyclic subgroup equals the order of its generator statement
  17. Cyclic groups of the same order are isomorphic statement
  18. Product of cyclic groups is cyclic iff the orders are coprime statement
  19. Classification of finite abelian groups statement
  20. A finite abelian group is cyclic iff its exponent equals its order statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Torsion_group, lead section
  2. Web https://en.wikipedia.org/wiki/Finitely_generated_abelian_group, Invariant factor decomposition
  3. Web https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n, Structure
About this page
Tag0MC0AQZSHE
URIm2://0MC0AQZSHE
Kindstatement, proposition
WritingW0 stub
MSC20K01
StatusThis copy of the site has no build data. Run m2 status 0MC0AQZSHE.
UsesExponent of a group, Cyclic group, Abelian group, Classification of finite abelian groups, Order of an element

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