Quotient-pullbackable equals inner: Difference between revisions

Latest revision as of 23:16, 16 September 2009

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

Statement

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

The automorphism is a quotient-pullbackable automorphism: For any homomorphism $ρ : H \to G$ , there is an automorphism $φ$ of $H$ , $ρ \circ φ = σ \circ ρ$ .
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 $ρ : H \to G$ there is an automorphism $φ$ of $H$ such that $ρ \circ φ = σ \circ ρ$ .

Inner automorphism

Further information: Inner automorphism

An automorphism $σ$ of a group $G$ is termed an inner automorphism if there exists $g \in G$ such that $σ = c_{g} = x \mapsto g x g^{- 1}$ .

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}

@@ Line 1: / Line 1: @@
-{{automorphism property implication|
+{{definition equivalence|inner automorphism}}
-stronger = quotient-pullbackable automorphism|
+{{factrelatedto|Extensible automorphisms problem}}
-weaker = inner automorphism}}
 ==Statement==
-Any [[quotient-pullbackable automorphism]] of a [[group]] is an [[inner automorphism]].
+The following are equivalent for an [[automorphism]] <math>\sigma</math> of a [[group]] <math>G</math>:
+# The automorphism is a quotient-pullbackable automorphism: For any homomorphism <math>\rho:H \to G</math>, there is an automorphism <math>\varphi</math> of <math>H</math>, <math>\rho \circ \varphi = \sigma \circ \rho</math>.
+# The automorphism is an [[inner automorphism]].
+==Definitions used==
+===Quotient-pullbackable automorphism===
+An automorphism <math>\sigma</math> of a group <math>G</math> is termed '''quotient-pullbackable''' if given any surjective homomorphism <math>\rho: H \to G</math> there is an automorphism <math>\varphi</math> of <math>H</math> such that <math>\rho \circ \varphi = \sigma \circ \rho</math>.
+===Inner automorphism===
+{{further|[[Inner automorphism]]}}
+An automorphism <math>\sigma</math> of a group <math>G</math> is termed an '''inner automorphism''' if there exists <math>g \in G</math> such that <math>\sigma = c_g = x \mapsto gxg^{-1}</math>.
 ==Related facts==