Finite-characteristic-semidirectly extensible implies linearly pushforwardable over prime field
This fact is related to: Extensible automorphisms problem
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Suppose is a finite group and is a prime not dividing the order of . Let be the prime field with elements. Then, any finite-characteristic-semidirectly extensible automorphism of is linearly pushforwardable over .
Finite-characteristic-semidirectly extensible automorphism
Further information: finite-characteristic-semidirectly extensible automorphism
An automorphism of is termed finite-characteristic-semidirectly extensible if, for any homomorphism where is a finite group, such that is a characteristic subgroup in the semidirect product , the automorphism extends to an automorphism of .
Linearly pushforwardable automorphism
Further information: Linearly pushforwardable automorphism An automorphism is termed linearly pushforwardable for a group over a field if, for any finite-dimensional linear representation , there exists such that for every , we have: