Math 2.0

example

The alternating group A3 is normal in S3

Example

In the symmetric group S3S_3, let A3={e,(1 2 3),(1 3 2)}A_3 = \{e, (1\,2\,3), (1\,3\,2)\}. This is the alternating group of degree 33. A3A_3 has 33 elements and S3S_3 has 66, so A3A_3 has index 22. The two left cosets are A3A_3 and the set of the three transpositions. The two right cosets are the same two sets. Thus gA3=A3ggA_3 = A_3g for each g∈S3g \in S_3, and A3A_3 is normal in S3S_3.

Remarks

This shows the result that subgroups of index two are normal.

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Alternating group, concept, no evidence levelAlternating groupconceptGroup, concept, no evidence levelGroupconceptSymmetric group, concept, no evidence levelSymmetric groupconceptSubgroup, definition, no evidence levelSubgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionIndex of a subgroup, definition, no evidence levelIndex of a subgroupdefinitionThe alternating group A3 is normal in S3, example, no evidence levelThe alternating group…example
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  1. Alternating group concept
  2. Group concept
  3. Symmetric group concept
  4. Subgroup definition
  5. Left coset definition
  6. Index of a subgroup definition
  7. The alternating group A3 is normal in S3 example

Sources

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

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