Characteristic-semidirectly extensible automorphism

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This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
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This is a variation of extensible automorphism|Find other variations of extensible automorphism |

This term is related to: Extensible automorphisms problem
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Definition

Definition with symbols

Let $\sigma$ be an automorphism of a group $G$ . We say that $\sigma$ is characteristic-semidirectly extensible or a CSE-automorphism if the following holds:

Let $\rho :G\to \operatorname {Aut} (N)$ be a homomorphism such that $N$ is a characteristic subgroup of the associated semidirect product $M=N\rtimes G$ . Then, there exists an automorphism $\phi$ of $N$ whose restriction to $G$ is $\sigma$ .

Relation with other proeprties

Stronger properties

Weaker properties

For a finite group, linearly pushforwardable automorphism over a prime field where the prime does not divide the order of the group. For full proof, refer: finite-characteristic-semidirectly extensible implies linearly pushforwardable over prime field