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Coprime automorphism-faithful characteristic subgroup

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A subgroup H of a finite group G is termed copime automorphism-faithful characteristic if every automorphism of G restricts to an automorphism of H (i.e., H is a characteristic subgroup) and if K is the kernel of the map:

\operatorname{Aut}(G) \to \operatorname{Aut}(H)

defined by restriction, then every prime divisor of the order of K, divides the order of G.

Relation with other properties