Math 2.0

Group theory and generalizations Example

The symmetric group S3 is not abelian

Builds on Symmetric groupTransposition

Example

Let S3S_3 be the symmetric group on {1,2,3}\{1, 2, 3\}. Compose permutations from right to left. Take the transpositions (1 2)(1\,2) and (1 3)(1\,3). Then (1 2)(1 3)=(1 3 2)(1\,2)(1\,3) = (1\,3\,2) and (1 3)(1 2)=(1 2 3)(1\,3)(1\,2) = (1\,2\,3). These two products are different. Thus S3S_3 is not abelian.

Remarks

S3S_3 has 66 elements. It is the smallest group that is not abelian.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Cycle notation, concept, no evidence levelCycle notationconceptGroup, concept, no evidence levelGroupconceptSymmetric group, concept, no evidence levelSymmetric groupconceptSymmetric group, definition, no evidence levelSymmetric groupdefinitionTransposition, definition, no evidence levelTranspositiondefinitionThe symmetric group S3 is not abelian, example, no evidence levelThe symmetric group S…example
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.
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  1. Cycle notation concept
  2. Group concept
  3. Symmetric group concept
  4. Symmetric group definition
  5. Transposition definition
  6. The symmetric group S3 is not abelian example

Further reading

  1. Web https://en.wikipedia.org/wiki/Dihedral_group_of_order_6, lead section
About this page
TagQSMVQ4EJ3D
URIm2://QSMVQ4EJ3D
Kindexample
WritingW0 stub
MSC20B30
Instance ofGroup
Not an instance ofAbelian group (note: two transpositions do not commute.), Abelian group (note: (1 2)(1 3) is not (1 3)(1 2).)
UsesSymmetric group, Transposition

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