Semidirectly extensible implies linearly pushforwardable for representation over prime field
From Groupprops
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Statement
Suppose F is a prime field (i.e., either a field of prime order or the field of rational numbers), and G is a group. Suppose V is a finite-dimensional vector space over F, and
be a linear representation of G. Let
with respect to the induced action of G on V.
Suppose, further, that σ is an automorphism of G that can be extended to an automorphism σ' of H such that σ' also restricts to an automorphism α of V. Then,
where cα is conjugation by α in GL(V).
Note that we need the field to be a prime field in order that GL(V) is equal to the automorphism group of V as a group.
Related facts
Applications
- 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
Facts used
- Automorphism group equals general linear group for vector space over prime field
- Automorphism group action lemma: Suppose H is a group, and
are subgroups such that
. Suppose σ' is an automorphism of H such that the restriction of σ' to N gives an automorphism α of N, and such that σ' also restricts to an automorphism of G, say σ. Consider the map:
that sends an element
to the automorphism of N induced by conjugation by g (note that this is an automorphism since
). Then, we have:
where cα denotes conjugation by α in the group
.
Proof
Given: A group G, a homomorphism
for a finite-dimensional vector space V over a prime field F. σ is an automorphism of G that extends to an automorphism σ' of H, such that σ' also restricts to an automorphism α of V.
To prove:
.
Proof: Since F is a prime field, GL(V) is the whole automorphism group of V by fact (1) (in general, it is a proper subgroup). Thus, the element α, which is a group automorphism of V, is actually in GL(V). Thus, fact (2), setting G = G,H = H,N = V,σ' = σ',α = α,σ = σ, gives the desired result.