Math 2.0

example

The symmetric group S3 is not abelian

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.

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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
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Sources

  1. Web https://en.wikipedia.org/wiki/Dihedral_group_of_order_6, lead section

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