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 copyMore info