# Finite-quotient-pullbackable implies class-preserving

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)
View all automorphism property implications | View all automorphism property non-implications

## 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.

## 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).