Math 2.0

Group theory and generalizations Proposition

Subgroups of abelian groups are normal

Builds on Normal subgroupAbelian group

Statement

Let GG be an abelian group and let HH be a subgroup of GG. Then HH is normal in GG.

Remarks

For g∈Gg \in G and h∈Hh \in H, commutativity gives ghg−1=hgg−1=h∈Hghg^{-1} = hgg^{-1} = h \in H. The converse is false: in the quaternion group Q8Q_8 each subgroup is normal, but Q8Q_8 is not abelian. More generally, each subgroup of the center of any group is normal.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptCommutativity, definition, no evidence levelCommutativitydefinitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionSubgroups of abelian groups are normal, statement, no evidence levelSubgroups of abelian …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. Binary operation definition
  2. Group concept
  3. Commutativity definition
  4. Conjugate elements definition
  5. Subgroup definition
  6. Abelian group definition
  7. Conjugate subgroup definition
  8. Normal subgroup definition
  9. Subgroups of abelian groups are normal statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Normal_subgroup, Examples
About this page
Tag0JPDTEDCFD
URIm2://0JPDTEDCFD
Kindstatement, proposition
WritingW0 stub
MSC20A05
StatusThis copy of the site has no build data. Run m2 status 0JPDTEDCFD.
UsesNormal subgroup, Abelian group

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