Quotient-pullbackable equals inner

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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 problemView terms related to Extensible automorphisms problem |

Statement

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

  1. The automorphism is a quotient-pullbackable automorphism: For any homomorphism \rho:H \to G, there is an automorphism \varphi of H, \rho \circ \varphi = \sigma \circ \rho.
  2. The automorphism is an inner automorphism.

Definitions used

Quotient-pullbackable automorphism

An automorphism \sigma of a group G is termed quotient-pullbackable if given any surjective homomorphism \rho: H \to G there is an automorphism \varphi of H such that \rho \circ \varphi = \sigma \circ \rho.

Inner automorphism

Further information: Inner automorphism

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

Related facts

References