Finite solvable-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)
View all subgroup property implications | View all subgroup property non-implications
Get more facts about Hall-extensible automorphism|Get more facts about class-preserving automorphism
Contents
Statement
Any finite solvable-extensible automorphism of a finite solvable group is a class-preserving automorphism.
Definitions used
Finite solvable-extensible automorphism
Further information: Finite solvable-extensible automorphism
An automorphism of a finite solvable group
is termed finite solvable-extensible if, for any finite solvable group
containing
, there is an automorphism
of
whose restriction to
equals
.
Class-preserving automorphism
Further information: Class-preserving automorphism
An automorphism of a finite group
is termed class-preserving if it sends every element to a conjugate element.
Related facts
Stronger facts
Other facts proved using the same method
- Hall-semidirectly extensible implies class-preserving
- Finite-extensible implies class-preserving
- Finite-quotient-pullbackable implies class-preserving
Facts used
- Finite solvable-extensible implies semidirectly extensible for representation over finite field of coprime characteristic
- Semidirectly extensible implies linearly pushforwardable for representation over prime field
- Linearly pushforwardable implies class-preesrving for class-separating field
- Every finite group admits a sufficiently large finite prime field
- Sufficiently large implies splitting, Splitting implies character-separating, Character-separating implies class-separating
Proof
Facts (1) and (2) combine to yield that any finite solvable-extensible automorphism is linearly pushforwardable over a finite 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).