Finite-characteristic-semidirectly extensible implies linearly pushforwardable over prime field

From Groupprops
Jump to: navigation, search
This fact is related to: Extensible automorphisms problem
View other facts related to Extensible automorphisms problemView terms related to Extensible automorphisms problem |

Statement

Suppose G is a finite group and p is a prime not dividing the order of G. Let k be the prime field with p elements. Then, any finite-characteristic-semidirectly extensible automorphism of G is linearly pushforwardable over k.

Definitions used

Finite-characteristic-semidirectly extensible automorphism

Further information: finite-characteristic-semidirectly extensible automorphism

An automorphism \sigma of G is termed finite-characteristic-semidirectly extensible if, for any homomorphism \rho:G \to \operatorname{Aut}(N) where N is a finite group, such that N is a characteristic subgroup in the semidirect product M = N \rtimes G, the automorphism \sigma extends to an automorphism of M.

Linearly pushforwardable automorphism

Further information: Linearly pushforwardable automorphism An automorphism \sigma is termed linearly pushforwardable for a group G over a field k if, for any finite-dimensional linear representation \rho:G \to GL(V), there exists a \in GL(V) such that for every g \in G, we have:

\rho(\sigma(g)) = a\rho(g)a^{-1}

Facts used

Proof

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]