Math 2.0

example

A subgroup of order two in S3 is not normal

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.

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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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Sources

  1. Web https://en.wikipedia.org/wiki/Normal_subgroup, Examples

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