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Class-preserving implies linearly extensible

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


The proof combines two facts: