Linearly pushforwardable automorphism

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Linearly pushforwardable automorphism, all facts related to Linearly pushforwardable automorphism) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

This article defines an automorphism property related to (or which arises in the context of): linear representation theory
View other automorphism properties related to linear representation theory OR View all terminology related to linear representation theory OR View all automorphism properties

This term is related to: Extensible automorphisms problem
View other terms related to Extensible automorphisms problem | View facts related to Extensible automorphisms problem

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:

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

Relation with other properties

Stronger properties

• Inner automorphism (unconditionally)
• Class-preserving automorphism when the field is a class-determining field (for instance, a field whose characteristic does not divide the order of the group, for a finite group)

Weaker properties

• Linearly extensible automorphism
• Class-preserving automorphism when the field is a class-separating field (for instance, a splitting field for a finite group)