Math 2.0

Group theory and generalizations Definition

Outer automorphism group

Builds on Automorphism groupInner automorphismQuotient group

Definition

Let GG be a group. The outer automorphism group of GG is the quotient group Out⁡(G)=Aut⁡(G)/Inn⁡(G)\operatorname{Out}(G) = \operatorname{Aut}(G) / \operatorname{Inn}(G).

Remarks

The quotient exists because Inn⁡(G)\operatorname{Inn}(G) is a normal subgroup of Aut⁡(G)\operatorname{Aut}(G). An automorphism that is not inner is called an outer automorphism, but the elements of Out⁡(G)\operatorname{Out}(G) are cosets, not automorphisms. For an abelian group, Inn⁡(G)\operatorname{Inn}(G) is trivial, so Out⁡(G)≅Aut⁡(G)\operatorname{Out}(G) \cong \operatorname{Aut}(G). The symmetric group S6S_6 is the only symmetric group SnS_n with nontrivial outer automorphism group.

Reading path

To understand this page from the ground up, read these 14 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 isomorphismdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionAutomorphism group, definition, no evidence levelAutomorphism groupdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionInner automorphism, definition, no evidence levelInner automorphismdefinitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementQuotient group, definition, no evidence levelQuotient groupdefinitionOuter automorphism group, definition, no evidence levelOuter automorphism gr…definition
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. Left coset definition
  8. Right coset definition
  9. Automorphism group definition
  10. Normal subgroup definition
  11. Inner automorphism definition
  12. Normality via left and right cosets statement
  13. Quotient group definition
  14. Outer automorphism group definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Outer_automorphism_group, lead section
About this page
Tag09HEXDQ0CQ
URIm2://09HEXDQ0CQ
Kinddefinition
WritingW1 stated
MSC20F28
DefinesOuter automorphism group
UsesAutomorphism group, Inner automorphism, Quotient group

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