Finite-quotient-pullbackable implies class-preserving

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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., finite-quotient-pullbackable automorphism) must also satisfy the second automorphism property (i.e., class-preserving automorphism)
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Statement

Suppose G is a finite group and \sigma is a finite-quotient-pullbackable automorphism of G. Then, \sigma is a class-preserving automorphism of G: it sends every element of G to within its conjugacy class.

Related facts

Facts used

  1. Finite-quotient-pullbackable implies quotient-pullbackable for representation over finite field
  2. Quotient-pullbackable implies linearly pushforwardable for representation over prime field
  3. Linearly pushforwardable implies class-preserving for class-separating field
  4. Every finite group admits a sufficiently large field
  5. Sufficiently large implies splitting, Splitting implies character-separating, Character-separating implies class-separating

Proof

By facts (1) and (2), any finite-quotient-pullbackable is linearly pushforwardable over any finite prime field. By fact (3), it suffices to show that there exists a finite prime field that is class-separating for the group. This is achieved by facts (4) and (5).