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This article gives a proof/explanation of the equivalence of multiple definitions for the term inner automorphism
View a complete list of pages giving proofs of equivalence of definitions

This fact is related to: Extensible automorphisms problem
View other facts related to Extensible automorphisms problem View terms related to Extensible automorphisms problem |

Contents 1 Statement 2 Definitions used 2.1 Quotient-pullbackable automorphism 2.2 Inner automorphism 3 Related facts 4 References

Statement

The following are equivalent for an automorphism σ of a group G:

1. The automorphism is a quotient-pullbackable automorphism: For any homomorphism , there is an automorphism of H, .
2. The automorphism is an inner automorphism.

Definitions used

Quotient-pullbackable automorphism

An automorphism σ of a group G is termed quotient-pullbackable if given any surjective homomorphism there is an automorphism of H such that .

Inner automorphism

Further information: Inner automorphism

An automorphism σ of a group G is termed an inner automorphism if there exists such that .

Related facts

• Extensible implies inner
• Finite-extensible implies inner
• Finite-quotient-pullbackable implies inner

References

• Characterizing inner automorphisms of groups by Martin R. Pettet, Archiv der Mathematik, ISSN 1420-8938 (Online), ISSN 0003-889X (Print), Volume 55,Number 5, Page 422 - 428(Year 1990): ^{Springerlink official copyMore info}