Pushforwardable equals inner

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.