Math 2.0

Group theory and generalizations Proposition

Product of a subgroup and a normal subgroup is a subgroup

Builds on Product of two subgroupsNormal subgroup

Statement

Let GG be a group, let HH be a subgroup of GG, and let NN be a normal subgroup of GG. Then the product HNHN is a subgroup of GG, and HN=NHHN = NH.

Remarks

If HH is also normal, then HNHN is normal in GG. The proof uses hN=NhhN = Nh for each h∈Hh \in H, so that HN=⋃h∈HhN=⋃h∈HNh=NHHN = \bigcup_{h \in H} hN = \bigcup_{h \in H} Nh = NH. This product appears in the second isomorphism theorem.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptConjugate elements, definition, no evidence levelConjugate elementsdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionProduct of two subgroups, definition, no evidence levelProduct of two subgro…definitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionProduct of a subgroup and a normal subgroup is a subgroup, statement, no evidence levelProduct of a subgroup…statement
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. Conjugate elements definition
  3. Subgroup definition
  4. Conjugate subgroup definition
  5. Product of two subgroups definition
  6. Normal subgroup definition
  7. Product of a subgroup and a normal subgroup is a subgroup statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Product_of_group_subsets, Product of subgroups
About this page
Tag0TXWV1JGG1
URIm2://0TXWV1JGG1
Kindstatement, proposition
WritingW0 stub
MSC20A05
StatusThis copy of the site has no build data. Run m2 status 0TXWV1JGG1.
UsesProduct of two subgroups, Normal subgroup

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