Semidirectly extensible implies linearly pushforwardable for representation over prime field

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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 \rho:G \to GL(V) be a linear representation of G. Let H = V \rtimes G with respect to the induced action of G on V.

Suppose, further, that \sigma is an automorphism of G that can be extended to an automorphism \sigma' of H such that \sigma' also restricts to an automorphism \alpha of V. Then, \rho \circ \sigma = c_\alpha \circ \rho where c_\alpha is conjugation by \alpha 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

Facts used

  1. Automorphism group equals general linear group for vector space over prime field
  2. Automorphism group action lemma: Suppose H is a group, and N,G \le H are subgroups such that G \le N_H(N). Suppose \sigma' is an automorphism of H such that the restriction of \sigma' to N gives an automorphism \alpha of N, and such that \sigma' also restricts to an automorphism of G, say \sigma. Consider the map:

\rho: G \to \operatorname{Aut}(N)

that sends an element g \in G to the automorphism of N induced by conjugation by g (note that this is an automorphism since G \le N_H(N)). Then, we have:

\rho \circ \sigma = c_\alpha \circ \rho

where c_\alpha denotes conjugation by \alpha in the group \operatorname{Aut}(N).

Proof

Given: A group G, a homomorphism \rho:G \to GL(V) for a finite-dimensional vector space V over a prime field F. \sigma is an automorphism of G that extends to an automorphism \sigma' of H, such that \sigma' also restricts to an automorphism \alpha of V.

To prove: \rho \circ \sigma = c_\alpha \circ \rho.

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 \alpha, which is a group automorphism of V, is actually in GL(V). Thus, fact (2), setting G = G, H = H, N = V, \sigma' = \sigma', \alpha = \alpha, \sigma = \sigma, gives the desired result.