Linearly extensible 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 extensible automorphism, all facts related to Linearly extensible 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 |

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 $g\in G$:

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

Relation with other properties

Stronger properties (subject to further conditions)

• Inner automorphism (unconditionally)
• Finite-extensible automorphism when the field is a prime field and the group is a finite group.
• Class-preserving automorphism when the field is class-determining for the group, which happens to be true in the non-modular case For full proof, refer: Class-preserving implies linearly extensible
• Galois-class automorphism in the non-modular case
• Linearly pushforwardable automorphism

Weaker properties (subject to further conditions)

• Class-preserving automorphism when the field is class-separating (for instance, when the group is finite and the field is a splitting field for the group) For full proof, refer: Linearly extensible implies class-preserving
• Galois-class automorphism in the non-modular case