Math 2.0

Group theory and generalizations Theorem

Number of elements of each order in a cyclic group

Builds on Cyclic groupOrder of a powerSubgroups of a finite cyclic group correspond to divisors of its order

Statement

Let GG be a cyclic group of finite order nn, and let dd be a positive divisor of nn. Then GG has exactly φ(d)\varphi(d) elements of order dd, where φ\varphi is Euler's totient function.

Remarks

Here φ(d)\varphi(d) is the number of integers kk with 1≤k≤d1 \leq k \leq d and gcd⁡(k,d)=1\gcd(k, d) = 1. The elements of order dd are the generators of the unique subgroup of order dd. Counting all elements by their orders gives the identity ∑d∣nφ(d)=n\sum_{d \mid n} \varphi(d) = n. In particular, GG has exactly φ(n)\varphi(n) generators.

Reading path

To understand this page from the ground up, read these 11 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…statementNumber of elements of each order in a cyclic group, statement, no evidence levelNumber of elements of…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
  11. Number of elements of each order in a cyclic group statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Cyclic_group, Additional properties
About this page
Tag0XCYGCCX8V
URIm2://0XCYGCCX8V
Kindstatement, theorem
WritingW0 stub
MSC20K01
StatusThis copy of the site has no build data. Run m2 status 0XCYGCCX8V.
UsesCyclic group, Order of a power, Subgroups of a finite cyclic group correspond to divisors of its order

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