Pushforwardable 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. \sigma is a pushforwardable automorphism of G: For any homomorphism \rho:G \to H, there exists an automorphism \varphi of H such that \rho \circ \sigma = \varphi \circ \rho.
  2. \sigma is an inner automorphism.

Definitions used

Pushforwardable automorphism

An automorphism \sigma of a group G is termed a pushforwardable automorphism if, for any homomorphism \rho:G \to H, there exists an automorphism \varphi of H such that \rho \circ \sigma = \varphi \circ \rho.

Inner automorphism

Further information: Inner automorphism

Facts used

  1. Extensible equals inner: An automorphism \sigma of a group G is inner if and only if it can be extended to an automorphism for any group containing G.

Proof

Proof that inner implies pushforwardable

If \sigma = c_g for g \in G, and \rho:G \to H is a homomorphism, we can take \varphi = c_{\sigma(g)}.

Proof that pushforwardable implies inner

If \sigma is pushforwardable, then, in particular, \sigma is extensible in the sense that it can be extended to an automorphism for any group containing G. Thus, by fact (1), \sigma is inner.