Finitely generated implies every subgroup of finite index has finitely many automorphic subgroups

Verbal statement
Any finitely generated group is a group in which every subgroup of finite index has finitely many automorphic subgroups.

Statement with symbols
The statement has the following equivalent formulation:


 * 1) Suppose $$H$$ is a subgroup of finite index in a finitely generated group $$G$$. Then, there are only finitely many subgroups $$K$$ of $$G$$ that are automorphic subgroups of $$H$$, i.e., for which there exists an automorphism $$\sigma$$ satisfying $$\sigma(H) = K$$.
 * 2) Suppose $$H$$ is a subgroup of finite index in a finitely generated group $$G$$. Then, the characteristic core of $$H$$ in $$G$$ is also a subgroup of finite index in $$G$$.
 * 3) Suppose $$H$$ is a subgroup of finite index in a finitely generated group $$G$$. Then, $$H$$ contains a characteristic subgroup of finite index in $$G$$.

Related facts

 * Poincare's theorem: This states that any subgroup of finite index contains a normal subgroup of finite index.
 * Finitely generated and residually finite implies Hopfian

Corollaries

 * Finitely generated implies every subgroup of finite index contains a characteristic subgroup of finite index

Facts used

 * 1) uses::Finitely generated implies finitely many homomorphisms to any finite group
 * 2) uses::Finitely many homomorphisms to any finite group implies every subgroup of finite index has finitely many automorphic subgroups