Math 2.0

example

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

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.

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To read this node, read these 11 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  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

Sources

  1. Web https://en.wikipedia.org/wiki/Quotient_group, Definition

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