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Finite-extensible implies semidirectly extensible for representation over finite field of coprime characteristic
From Groupprops
Statement
Suppose G is a finite group and σ is a finite-extensible automorphism of G. Suppose ρ 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 
be the semidirect product corresponding to the representation. Then, σ extends to an automorphism σ' of H such that σ' also restricts to an automorphism of V.
Proof
Given: A finite group G, a finite-extensible automorphism σ of G, a representation of G over a finite field of characteristic p that does not divide the order of G. is the corresponding semidirect product.
To prove: σ extends to an automorphism σ' of H that also restricts to an automorphism of V.
Proof:
- Since V is a finite-dimensional vector space over a finite field, it is finite, and hence is a finite group containing G.
- Since σ is finite-extensible, it extends to an automorphism σ' of H.
- 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, σ' restricts to an automorphism of V.
