Math 2.0

Group theory and generalizations Theorem

Inner automorphism group is isomorphic to G modulo its centre

Builds on Inner automorphismCentre of a groupFirst isomorphism theorem

Statement

Let GG be a group with centre Z(G)Z(G). Then the map gZ(G)↦cggZ(G) \mapsto c_g, where cg(x)=gxg−1c_g(x) = gxg^{-1}, is an isomorphism from G/Z(G)G/Z(G) to Inn⁡(G)\operatorname{Inn}(G).

Remarks

The map g↦cgg \mapsto c_g is a surjective homomorphism from GG to Inn⁡(G)\operatorname{Inn}(G). Its kernel is Z(G)Z(G), because cgc_g is the identity exactly when gg commutes with each element. The first isomorphism theorem then gives the result. For example, the symmetric group S3S_3 has trivial centre, so Inn⁡(S3)≅S3\operatorname{Inn}(S_3) \cong S_3.

Reading path

To understand this page from the ground up, read these 32 pages in order. The foundations come first. The path takes the proofs Proof that the kernel is a normal subgroup, Proof that a homomorphism is injective iff its kernel is trivial, Proof of the first isomorphism theorem, Proof of the uniqueness of inverses, Proof of the cancellation laws.

Show the path as a graph
Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptIdentity element, definition, no evidence levelIdentity elementdefinitionCentre of a group, definition, no evidence levelCentre of a groupdefinitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionInverse element, definition, no evidence levelInverse elementdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionGroup isomorphism, definition, no evidence levelGroup isomorphismdefinitionKernel of a homomorphism, definition, no evidence levelKernel of a homomorph…definitionImage of a homomorphism, definition, no evidence levelImage of a homomorphi…definitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionProof of the cancellation laws, proof, no evidence levelProof of the cancella…proofProof of the uniqueness of inverses, proof, no evidence levelProof of the uniquene…proofAutomorphism group, definition, no evidence levelAutomorphism groupdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionCancellation laws, statement, no evidence levelCancellation lawsstatementUniqueness of inverses, statement, no evidence levelUniqueness of inversesstatementInner automorphism, definition, no evidence levelInner automorphismdefinitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementHomomorphisms preserve identity and inverses, statement, no evidence levelHomomorphisms preserv…statementQuotient group, definition, no evidence levelQuotient groupdefinitionProof that a homomorphism is injective iff its kernel is trivial, proof, no evidence levelProof that a homomorp…proofProof that the kernel is a normal subgroup, proof, no evidence levelProof that the kernel…proofUniversal property of the quotient group, statement, no evidence levelUniversal property of…statementA homomorphism is injective iff its kernel is trivial, statement, no evidence levelA homomorphism is inj…statementThe kernel is a normal subgroup, statement, no evidence levelThe kernel is a norma…statementProof of the first isomorphism theorem, proof, no evidence levelProof of the first is…proofFirst isomorphism theorem, statement, no evidence levelFirst isomorphism the…statementInner automorphism group is isomorphic to G modulo its centre, statement, no evidence levelInner automorphism gr…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. Identity element definition
  4. Centre of a group definition
  5. Conjugate elements definition
  6. Group homomorphism definition
  7. Subgroup definition
  8. Inverse element definition
  9. Conjugate subgroup definition
  10. Group isomorphism definition
  11. Kernel of a homomorphism definition
  12. Image of a homomorphism definition
  13. Left coset definition
  14. Right coset definition
  15. Proof of the cancellation laws proof
  16. Proof of the uniqueness of inverses proof
  17. Automorphism group definition
  18. Normal subgroup definition
  19. Cancellation laws statement
  20. Uniqueness of inverses statement
  21. Inner automorphism definition
  22. Normality via left and right cosets statement
  23. Homomorphisms preserve identity and inverses statement
  24. Quotient group definition
  25. Proof that a homomorphism is injective iff its kernel is trivial proof
  26. Proof that the kernel is a normal subgroup proof
  27. Universal property of the quotient group statement
  28. A homomorphism is injective iff its kernel is trivial statement
  29. The kernel is a normal subgroup statement
  30. Proof of the first isomorphism theorem proof
  31. First isomorphism theorem statement
  32. Inner automorphism group is isomorphic to G modulo its centre statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Inner_automorphism, Inner and outer automorphism groups
About this page
Tag05HYGWFG63
URIm2://05HYGWFG63
Kindstatement, theorem
WritingW0 stub
MSC20F28
StatusThis copy of the site has no build data. Run m2 status 05HYGWFG63.
UsesInner automorphism, Centre of a group, First isomorphism theorem

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