# Hall-extensible implies class-preserving

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

### Hall-extensible automorphism

Further information: Hall-extensible automorphism

An automorphism $\sigma$ of a finite group $G$ is termed Hall-extensible if, for any group $H$ containing $G$ as a Hall subgroup, there is an automorphism $\sigma'$ of $H$ whose restriction to $G$ equals $\sigma$.

### Class-preserving automorphism

Further information: Class-preserving automorphism

An automorphism $\sigma$ of a finite group $G$ is termed class-preserving if it sends every element to a conjugate element.

## Proof

Facts (1) and (2) combine to yield that any Hall-extensible automorphism is linearly pushforwardable over a prime field where the prime does not divide the order of the group, and fact (3) yields that if the field chosen is a class-separating field for the group, then the automorphism is class-preserving. Thus, we need to show that for every finite group, there exists a prime field with the prime not dividing the order of the group, such that the field is a class-separating field for the group. This is achieved by facts (4) and (5).