Class-preserving implies linearly extensible

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This article gives the statement and possibly, proof, of an implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., class-preserving automorphism) must also satisfy the second automorphism property (i.e., linearly extensible automorphism)
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Definition

Let G be a group, and k a class-determining field for it. Then, any class-preserving automorphism of G is linearly extensible.

Proof

The proof combines two facts: