Quotient-pullbackable equals inner
From Groupprops
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 problemView terms related to Extensible automorphisms problem |
Contents
Statement
The following are equivalent for an automorphism of a group :
- The automorphism is a quotient-pullbackable automorphism: For any homomorphism , there is an automorphism of , .
- The automorphism is an inner automorphism.
Definitions used
Quotient-pullbackable automorphism
An automorphism of a group is termed quotient-pullbackable if given any surjective homomorphism there is an automorphism of such that .
Inner automorphism
Further information: Inner automorphism
An automorphism of a group is termed an inner automorphism if there exists such that .
Related facts
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 copy}^{More info}