Linearly extensible automorphism

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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 |

History

Origin

The notion arose through attempts to use representation theory in solving the Extensible automorphisms problem.

Definition

Symbol-free definition

Given a group and a field, an automorphism of the group is termed linearly extensible if it extends to an inner automorphism (of the general linear group)for every faithful finite-dimensional linear representation of the group over the field.

Definition with symbols

Given a group $G$ and a field $k$ an automorphism $\sigma$ of $G$ is termed linearly extensible if for any faithful linear representation $\varphi: G \to GL_n(k)$, there exists $a \in GL_n(k)$ such that for any $g \in G$:

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