Finite-extensible implies semidirectly extensible for representation over finite field of coprime characteristic

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Statement

Suppose G is a finite group and \sigma is a finite-extensible automorphism of G. Suppose \rho is a linear representation of G over a finite field whose characteristic does not divide the order of G. Let V be the corresponding vector space and H = V \rtimes G be the semidirect product corresponding to the representation. Then, \sigma extends to an automorphism \sigma' of H such that \sigma' also restricts to an automorphism of V.

Proof

Given: A finite group G, a finite-extensible automorphism \sigma of G, a representation of G over a finite field of characteristic p that does not divide the order of G. H = V \rtimes G is the corresponding semidirect product.

To prove: \sigma extends to an automorphism \sigma' of H that also restricts to an automorphism of V.

Proof:

  1. Since V is a finite-dimensional vector space over a finite field, it is finite, and hence H = V \rtimes G is a finite group containing G.
  2. Since \sigma is finite-extensible, it extends to an automorphism \sigma' of H.
  3. Finally, V is a normal p-Sylow subgroup of H, and hence, characteristic in H (it can be described as the set of elements whose order divides p). Thus, \sigma' restricts to an automorphism of V.