Finite-characteristic-semidirectly extensible implies linearly pushforwardable over prime field

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

Statement

Suppose $G$ is a finite group and $p$ is a prime not dividing the order of $G$. Let $k$ be the prime field with $p$ elements. Then, any finite-characteristic-semidirectly extensible automorphism of $G$ is linearly pushforwardable over $k$.

Definitions used

Finite-characteristic-semidirectly extensible automorphism

Further information: finite-characteristic-semidirectly extensible automorphism

An automorphism $\sigma$ of $G$ is termed finite-characteristic-semidirectly extensible if, for any homomorphism $\rho :G\to \operatorname {Aut} (N)$ where $N$ is a finite group, such that $N$ is a characteristic subgroup in the semidirect product $M=N\rtimes G$, the automorphism $\sigma$ extends to an automorphism of $M$.

Linearly pushforwardable automorphism

Further information: Linearly pushforwardable automorphism An automorphism $\sigma$ is termed linearly pushforwardable for a group $G$ over a field $k$ if, for any finite-dimensional linear representation $\rho :G\to GL(V)$, there exists $a\in GL(V)$ such that for every $g\in G$, we have:

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

Facts used

• Automorphism group action lemma

Proof

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