# Linearly pushforwardable automorphism

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines an automorphism property related to (or which arises in the context of): linear representation theory
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This term is related to: Extensible automorphisms problem
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This is a variation of extensible automorphism|Find other variations of extensible automorphism |

## Definition

### Definition with symbols

Let $G$ be a group and $k$ be a field. An automorphism $\sigma$ of $G$ is termed linearly pushforwardable if for any linear representation $\varphi:G \to GL_n(k)$, there exists $a \in GL_n(k)$ such that for any $g \in G$, we have:

$\varphi(\sigma(g)) = a\varphi(g)a^{-1}$