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

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finitely generated group. That is, it states that in a Finitely generated group (?), every subgroup satisfying the first subgroup property (i.e., Subgroup of finite index (?)) must also satisfy the second subgroup property (i.e., Subgroup having finitely many automorphic subgroups (?)). In other words, every subgroup of finite index of finitely generated group is a subgroup having finitely many automorphic subgroups of finitely generated group.
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This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finitely generated group) must also satisfy the second group property (i.e., group in which every subgroup of finite index has finitely many automorphic subgroups)
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## Statement

### 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 $G$.