Math 2.0

Group theory and generalizations Proposition

Characteristic subgroups are normal

Builds on Characteristic subgroupInner automorphismNormal subgroup

Statement

Let GG be a group and let HH be a characteristic subgroup of GG. Then HH is a normal subgroup of GG.

Remarks

Each inner automorphism cgc_g is an automorphism, so gHg−1=cg(H)=HgHg^{-1} = c_g(H) = H for each g∈Gg \in G. The converse is false: in the Klein four-group V={e,a,b,ab}V = \{e, a, b, ab\}, each subgroup is normal, but an automorphism can swap {e,a}\{e, a\} and {e,b}\{e, b\}.

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 levelGroupconceptConjugate elements, definition, no evidence levelConjugate elementsdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionGroup isomorphism, definition, no evidence levelGroup isomorphismdefinitionAutomorphism group, definition, no evidence levelAutomorphism groupdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionInner automorphism, definition, no evidence levelInner automorphismdefinitionCharacteristic subgroup, definition, no evidence levelCharacteristic subgro…definitionCharacteristic subgroups are normal, statement, no evidence levelCharacteristic subgro…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. Group homomorphism definition
  4. Subgroup definition
  5. Conjugate subgroup definition
  6. Group isomorphism definition
  7. Automorphism group definition
  8. Normal subgroup definition
  9. Inner automorphism definition
  10. Characteristic subgroup definition
  11. Characteristic subgroups are normal statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Characteristic_subgroup, Normal subgroup
About this page
Tag0SRE7K4E49
URIm2://0SRE7K4E49
Kindstatement, proposition
WritingW0 stub
MSC20F28
StatusThis copy of the site has no build data. Run m2 status 0SRE7K4E49.
UsesCharacteristic subgroup, Inner automorphism, Normal subgroup

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