Math 2.0

proof

Proof that subgroups of index two are normal

Proof

  1. Let HH be a subgroup of a group GG with index 22.
  2. There are two left cosets. One is HH, so the other is G∖HG \setminus H.
  3. In the same way, there are two right cosets, HH and G∖HG \setminus H.
  4. Let g∈Gg \in G. If g∈Hg \in H, then gH=H=HggH = H = Hg.
  5. If g∉Hg \notin H, then gH≠HgH \ne H and Hg≠HHg \ne H. Thus gH=G∖H=HggH = G \setminus H = Hg.
  6. So gH=HggH = Hg for all g∈Gg \in G, and HH is normal.

Idea

With only two cosets, one of them is the subgroup and the other is its complement. This is true on the left and on the right, so the left and right cosets agree.

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Group, concept, no evidence levelGroupconceptLeft coset, concept, no evidence levelLeft cosetconceptConjugate 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 subgroupdefinitionIndex of a subgroup, definition, no evidence levelIndex of a subgroupdefinitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementProof that subgroups of index two are normal, proof, no evidence levelProof that subgroups …proof
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.
  • no evidence level
  1. Group concept
  2. Left coset 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. Index of a subgroup definition
  10. Normality via left and right cosets statement
  11. Proof that subgroups of index two are normal proof

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  1. Web https://en.wikipedia.org/wiki/Index_of_a_subgroup, Properties

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