Math 2.0

Group theory and generalizations Proposition

Order of a cyclic subgroup equals the order of its generator

Builds on Cyclic subgroup generated by an elementOrder of an elementOrder of a groupg to the n equals 1 iff the order divides n

Statement

Let GG be a group and let g∈Gg \in G. If gg has finite order nn, then ⟨g⟩={e,g,g2,…,gn−1}\langle g \rangle = \{e, g, g^2, \ldots, g^{n-1}\}, these nn elements are distinct, and thus ∣⟨g⟩∣=n|\langle g \rangle| = n. If gg has infinite order, then the powers gkg^k for k∈ℤk \in \mathbb{Z} are distinct, and ⟨g⟩\langle g \rangle is infinite.

Remarks

Here ⟨g⟩\langle g \rangle is the cyclic subgroup generated by gg, and ∣⟨g⟩∣|\langle g \rangle| is its order. The proof uses the divisibility criterion for powers: ga=gbg^a = g^b exactly when nn divides a−ba - b. With Lagrange's theorem, this fact shows that the order of an element of a finite group divides the order of the group.

Reading path

To understand this page from the ground up, read these 8 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 …statementOrder of a group, definition, no evidence levelOrder of a groupdefinitionSubgroup, 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…statementOrder of a cyclic subgroup equals the order of its generator, statement, no evidence levelOrder of a cyclic sub…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. Order of a group definition
  4. Subgroup definition
  5. Order of an element definition
  6. Cyclic subgroup generated by an element definition
  7. g to the n equals 1 iff the order divides n statement
  8. Order of a cyclic subgroup equals the order of its generator statement

Where this leads

16 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cyclic_group, Definition and notation
  2. Web https://en.wikipedia.org/wiki/Order_(group_theory), Order and structure
About this page
Tag031M8HX7DA
URIm2://031M8HX7DA
Kindstatement, proposition
WritingW0 stub
MSC20A05
StatusThis copy of the site has no build data. Run m2 status 031M8HX7DA.
UsesCyclic subgroup generated by an element, Order of an element, Order of a group, g to the n equals 1 iff the order divides n

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