Math 2.0

Group theory and generalizations Proof

Proof that subgroups of index two are normal

Builds on Index of a subgroupNormality via left and right cosetsLeft coset

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.

Reading path

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

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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 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. 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

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Index_of_a_subgroup, Properties
About this page
Tag4N1WKH2ZQV
URIm2://4N1WKH2ZQV
Kindproof
WritingW0 stub
MSC20A05
ProvesSubgroups of index two are normal
UsesIndex of a subgroup, Normality via left and right cosets, Left coset
MethodsProof by cases

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