Finite-extensible implies semidirectly extensible for representation over finite field of coprime characteristic
Suppose is a finite group and is a finite-extensible automorphism of . Suppose is a linear representation of over a finite field whose characteristic does not divide the order of . Let be the corresponding vector space and be the semidirect product corresponding to the representation. Then, extends to an automorphism of such that also restricts to an automorphism of .
Given: A finite group , a finite-extensible automorphism of , a representation of over a finite field of characteristic that does not divide the order of . is the corresponding semidirect product.
To prove: extends to an automorphism of that also restricts to an automorphism of .
- Since is a finite-dimensional vector space over a finite field, it is finite, and hence is a finite group containing .
- Since is finite-extensible, it extends to an automorphism of .
- Finally, is a normal -Sylow subgroup of , and hence, characteristic in (it can be described as the set of elements whose order divides ). Thus, restricts to an automorphism of .