Math 2.0

Group theory and generalizations Proof

Proof that cyclic groups are abelian

Builds on Cyclic groupLaws of exponents in a groupAbelian group

Proof

  1. Let GG be a cyclic group with generator gg.
  2. Let xx and yy be elements of GG. Then there are integers mm and nn with x=gmx = g^m and y=gny = g^n.
  3. By the laws of exponents, xy=gmgn=gm+nxy = g^m g^n = g^{m+n}.
  4. In the same way, yx=gngm=gn+myx = g^n g^m = g^{n+m}.
  5. Since m+n=n+mm + n = n + m, xy=yxxy = yx. Thus GG is abelian.

Idea

Every element is a power of one generator, and powers of the same element commute.

Reading path

To understand this page from the ground up, read these 9 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 levelCommutativitydefinitionLaws of exponents in a group, statement, no evidence levelLaws of exponents in …statementSubgroup, definition, no evidence levelSubgroupdefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionCyclic subgroup generated by an element, definition, no evidence levelCyclic subgroup gener…definitionCyclic group, definition, no evidence levelCyclic groupdefinitionProof that cyclic groups are abelian, proof, no evidence levelProof that cyclic gro…proof
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. Laws of exponents in a group statement
  5. Subgroup definition
  6. Abelian group definition
  7. Cyclic subgroup generated by an element definition
  8. Cyclic group definition
  9. Proof that cyclic groups are abelian proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Group_(mathematics), Cyclic groups
About this page
Tag8MADRXG8PQ
URIm2://8MADRXG8PQ
Kindproof
WritingW0 stub
MSC20A05
ProvesCyclic groups are abelian
UsesCyclic group, Laws of exponents in a group, Abelian group
MethodsDirect proof

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