Math 2.0

statement proposition

Characteristic subgroups are normal

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\}.

Explanations

L1 human

Normal means fixed by the inner automorphisms, and characteristic means fixed by all automorphisms. All automorphisms include the inner ones. So a characteristic subgroup is normal.

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To read this node, read these 11 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  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

Sources

  1. Web https://en.wikipedia.org/wiki/Characteristic_subgroup, Normal subgroup

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