Math 2.0

statement theorem

Inner automorphism group is isomorphic to G modulo its centre

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.

Explanations

L1 human

Two elements give the same conjugation exactly when they differ by an element of the centre. So the inner automorphisms are counted by the cosets of the centre. A group with a big centre has few inner automorphisms.

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To read this node, read these 26 nodes in this order. The foundations come first, and the list ends with this node.

Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptIdentity element, definition, no evidence levelIdentity elementdefinitionCancellation laws, statement, no evidence levelCancellation lawsstatementCentre 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 cosetdefinitionUniqueness of inverses, statement, no evidence levelUniqueness of inversesstatementAutomorphism group, definition, no evidence levelAutomorphism groupdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionHomomorphisms preserve identity and inverses, statement, no evidence levelHomomorphisms preserv…statementInner automorphism, definition, no evidence levelInner automorphismdefinitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementThe kernel is a normal subgroup, statement, no evidence levelThe kernel is a norma…statementQuotient group, definition, no evidence levelQuotient groupdefinitionUniversal property of the quotient group, statement, no evidence levelUniversal property of…statementFirst 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 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. Binary operation definition
  2. Group concept
  3. Identity element definition
  4. Cancellation laws statement
  5. Centre of a group definition
  6. Conjugate elements definition
  7. Group homomorphism definition
  8. Subgroup definition
  9. Inverse element definition
  10. Conjugate subgroup definition
  11. Group isomorphism definition
  12. Kernel of a homomorphism definition
  13. Image of a homomorphism definition
  14. Left coset definition
  15. Right coset definition
  16. Uniqueness of inverses statement
  17. Automorphism group definition
  18. Normal subgroup definition
  19. Homomorphisms preserve identity and inverses statement
  20. Inner automorphism definition
  21. Normality via left and right cosets statement
  22. The kernel is a normal subgroup statement
  23. Quotient group definition
  24. Universal property of the quotient group statement
  25. First isomorphism theorem statement
  26. Inner automorphism group is isomorphic to G modulo its centre statement

Sources

  1. Web https://en.wikipedia.org/wiki/Inner_automorphism, Inner and outer automorphism groups

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