Math 2.0

Group theory and generalizations Example

A subgroup of order two in S3 is not normal

Builds on Symmetric groupLeft coset

Example

In the symmetric group S3S_3, let H={e,(1 2)}H = \{e, (1\,2)\}. Compose permutations from right to left. HH is a subgroup, because (1 2)(1 2)=e(1\,2)(1\,2) = e. The left coset of (1 2 3)(1\,2\,3) is (1 2 3)H={(1 2 3),(1 3)}(1\,2\,3)H = \{(1\,2\,3), (1\,3)\}. The right coset of (1 2 3)(1\,2\,3) is H(1 2 3)={(1 2 3),(2 3)}H(1\,2\,3) = \{(1\,2\,3), (2\,3)\}. These cosets are different. Thus HH is not normal in S3S_3.

Remarks

HH has index 33. Thus the result for index two does not apply to HH.

Reading path

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Group, concept, no evidence levelGroupconceptSymmetric group, concept, no evidence levelSymmetric groupconceptSubgroup, definition, no evidence levelSubgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionA subgroup of order two in S3 is not normal, example, no evidence levelA subgroup of order t…example
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  1. Group concept
  2. Symmetric group concept
  3. Subgroup definition
  4. Left coset definition
  5. A subgroup of order two in S3 is not normal example

Further reading

  1. Web https://en.wikipedia.org/wiki/Normal_subgroup, Examples
About this page
TagXD0WBC1MWB
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MSC20B30
Instance ofSubgroup
Not an instance ofNormal subgroup (note: a left coset and a right coset differ.), Normal subgroup (note: (1 2 3)H is not H(1 2 3).)
UsesSymmetric group, Left coset

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