Math 2.0

Group theory and generalizations Definition

Characteristic subgroup

Builds on Automorphism groupSubgroup

Definition

Let GG be a group and let HH be a subgroup of GG. The subgroup HH is characteristic in GG when α(H)=H\alpha(H) = H for each automorphism α\alpha of GG.

Remarks

It is equivalent to ask α(H)⊆H\alpha(H) \subseteq H for each automorphism α\alpha, because then also α−1(H)⊆H\alpha^{-1}(H) \subseteq H. The centre and the commutator subgroup of a group are characteristic. A characteristic subgroup of a normal subgroup of GG is normal in GG.

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Group, concept, no evidence levelGroupconceptGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionGroup isomorphism, definition, no evidence levelGroup isomorphismdefinitionAutomorphism group, definition, no evidence levelAutomorphism groupdefinitionCharacteristic subgroup, definition, no evidence levelCharacteristic subgro…definition
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  1. Group concept
  2. Group homomorphism definition
  3. Subgroup definition
  4. Group isomorphism definition
  5. Automorphism group definition
  6. Characteristic subgroup definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Characteristic_subgroup, Definition
About this page
Tag01CS8Q2APC
URIm2://01CS8Q2APC
Kinddefinition
WritingW1 stated
MSC20F28
DefinesCharacteristic subgroup
UsesAutomorphism group, Subgroup

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