Math 2.0

Group theory and generalizations Example

The cosets of a subgroup that is not normal do not form a group

Builds on Symmetric groupLeft cosetQuotient groupNormal subgroup

Example

In the symmetric group S3S_3, let H={e,(1 2)}H = \{e, (1\,2)\}. Compose permutations from right to left. Two of the left cosets are (1 3)H={(1 3),(1 2 3)}(1\,3)H = \{(1\,3), (1\,2\,3)\} and (2 3)H={(2 3),(1 3 2)}(2\,3)H = \{(2\,3), (1\,3\,2)\}. Try to set (aH)(bH)=(ab)H(aH)(bH) = (ab)H. With a=(1 3)a = (1\,3) and b=(2 3)b = (2\,3), we get ab=(1 3 2)ab = (1\,3\,2), which is in (2 3)H(2\,3)H. With a=(1 2 3)a = (1\,2\,3) from the same coset, we get ab=(1 2)ab = (1\,2), which is in HH. The two results are in different cosets. Thus this product is not defined, and the cosets of HH do not form a quotient group.

Remarks

HH is not normal in S3S_3. For a normal subgroup the product of cosets is always defined.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptSymmetric group, concept, no evidence levelSymmetric groupconceptConjugate elements, definition, no evidence levelConjugate elementsdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementQuotient group, definition, no evidence levelQuotient groupdefinitionThe cosets of a subgroup that is not normal do not form a group, example, no evidence levelThe cosets of a subgr…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.
  • no evidence level
  1. Group concept
  2. Symmetric group concept
  3. Conjugate elements definition
  4. Subgroup definition
  5. Conjugate subgroup definition
  6. Left coset definition
  7. Right coset definition
  8. Normal subgroup definition
  9. Normality via left and right cosets statement
  10. Quotient group definition
  11. The cosets of a subgroup that is not normal do not form a group example

Further reading

  1. Web https://en.wikipedia.org/wiki/Quotient_group, Definition
About this page
TagN5NTPAR2KT
URIm2://N5NTPAR2KT
Kindexample
WritingW0 stub
MSC20B30
Not an instance ofQuotient group (note: the product of cosets depends on the choice of representatives.)
UsesSymmetric group, Left coset, Quotient group, Normal subgroup

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