Groupprops, The Group Properties Wiki (pre-alpha)
YOUR FEEDBACK IS IMPORTANT!
Please take a short user satisfaction survey about Groupprops.
Your survey responses will be helpful in improving the site experience!
Thanks in advance!
Semidirectly extensible implies linearly pushforwardable for representation over prime field
From Groupprops
Contents |
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.

