Semidirectly extensible implies linearly pushforwardable for representation over prime field
Suppose is a prime field (i.e., either a field of prime order or the field of rational numbers), and is a group. Suppose is a finite-dimensional vector space over , and be a linear representation of . Let with respect to the induced action of on .
Suppose, further, that is an automorphism of that can be extended to an automorphism of such that also restricts to an automorphism of . Then, where is conjugation by in .
Note that we need the field to be a prime field in order that is equal to the automorphism group of as a group.
- Finite-extensible implies class-preserving
- Hall-semidirectly extensible implies class-preserving
- Finite solvable-extensible implies class-preserving
- Conjugacy-separable with only finitely many prime divisors of orders of elements implies every extensible automorphism is class-preserving
- Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving
- Automorphism group equals general linear group for vector space over prime field
- Automorphism group action lemma: Suppose is a group, and are subgroups such that . Suppose is an automorphism of such that the restriction of to gives an automorphism of , and such that also restricts to an automorphism of , say . Consider the map:
that sends an element to the automorphism of induced by conjugation by (note that this is an automorphism since ). Then, we have:
where denotes conjugation by in the group .
Given: A group , a homomorphism for a finite-dimensional vector space over a prime field . is an automorphism of that extends to an automorphism of , such that also restricts to an automorphism of .
To prove: .
Proof: Since is a prime field, is the whole automorphism group of by fact (1) (in general, it is a proper subgroup). Thus, the element , which is a group automorphism of , is actually in . Thus, fact (2), setting , gives the desired result.