# Conjugacy-separable with only finitely many prime divisors of orders of elements implies every extensible automorphism is class-preserving

From Groupprops

## Contents

## Statement

Suppose is a Conjugacy-separable group (?): in other words, given any two elements of that are not conjugate, there exists a normal subgroup of finite index in such that the images of in are not conjugate in .

Suppose, further, that the set of primes that divide the order of some non-identity element of is finite.

Then, if is an Extensible automorphism (?) of , is a Class-preserving automorphism (?) of .

## Related facts

- Finite-extensible implies class-preserving
- Conjugacy-separable implies every quotient-pullbackable automorphism is class-preserving
- Conjugacy-separable implies every semidirectly extensible automorphism is class-preserving
- Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving

## Facts used

- Semidirectly extensible implies linearly pushforwardable for representation over prime field
- Every finite group admits infinitely many sufficiently large prime fields
- Sufficiently large implies splitting, Splitting implies character-separating, Character-separating implies class-separating

## Proof

**Given**: A conjugacy-separable group with only finitely many primes dividing the orders of elements of . An extensible automorphism of . Two elements of that are not conjugate.

**To prove**: cannot send to .

**Proof**:

- There exists a normal subgroup of finite index in such that the images of and are not conjugate in : This follows from the definition of conjugacy-separable.
- Let . Then, there exists a prime such that the field of elements is sufficiently large for , and such that does not divide the order of any element of : By fact (2), there are infinitely many sufficiently large prime fields for , i.e., there are infinitely many primes for which the corresponding prime field is sufficiently large for . Since there are only finitely many prime divisors of orders of elements, we can find a prime
*not*among any of these divisors such that the corresponding prime field is sufficiently large. - The field of elements is a class-separating field for . In particular, there is a finite-dimensional linear representation of over this field such that and are not conjugate: This follows from fact (3).
- is linearly pushforwardable over the prime field with elements, for the chosen above. In particular, if , then and are conjugate for any representation over this field: Let be a representation of over this field. Let be the corresponding vector space and the semidirect product for the action. Since does not divide the order of any element of , is the set of elements of of order dividing . In particular, is characteristic in , and thus, if extends to an automorphism of , then also restricts to an automorphism of . Fact (1) thus yields that , so is linearly pushforwardable over the field of elements. In particular, if , then , so is conjugate to by .
- Let be the linear representation chosen in step (3), and let where is the quotient map. Then, and are not conjugate in the general linear group . However, by step (4), we have that is linearly pushforwardable, so if , then and are conjugate. This gives a contradiction, so we cannot have , and we are done.