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
View other terms related to Extensible automorphisms problem | View facts related to Extensible automorphisms problem

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