Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving
This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., conjugacy-separable aperiodic group) must also satisfy the second group property (i.e., group in which every extensible automorphism is class-preserving)
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Suppose is a Conjugacy-separable group (?) (i.e., any two non-conjugate elements can be realized as non-conjugate elements in some finite quotient) that is also an Aperiodic group (?): no non-identity element has finite order. Then, every Extensible automorphism (?) of is a Class-preserving automorphism (?).
- Conjugacy-separable and only finitely many prime divisors of orders of elements implies every extensible automorphism is class-preserving
- 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
- Semidirectly extensible implies linearly pushforwardable for representation over prime field
- Every finite group admits a sufficiently large finite prime field
- Sufficiently large implies splitting, Splitting implies character-separating, Character-separating implies class-separating
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 .
- 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 : This follows from fact (2).
- 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.