# Group in which every subgroup of finite index has finitely many automorphic subgroups

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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## Definition

### Equivalent definition in tabular format

No. Shorthand A group $G$ is a group in which every subgroup of finite index has finitely many automorphic subgroups if ...
1 finitely many automorphic subgroups for every subgroup of finite index $H$ in $G$, there are only finitely many automorphic subgroups to $H$ in $G$, i.e., there are only finitely many subgroups $K$ of $G$ for which there exists an automorphism $\sigma$ of $G$ such that $\sigma(H) = K$.
2 characteristic core has finite index for every subgroup of finite index $H$ in $G$, the characteristic core of $H$ in $G$ has finite index in $G$.
3 contains characteristic subgroup of finite index for every subgroup of finite index $H$ in $G$, there exists a characteristic subgroup of finite index $L$ in $G$ such that $L$ is contained in $H$.
4 normal subgroup, finitely many automorphic subgroups for every normal subgroup of finite index $H$ in $G$, there are only finitely many automorphic subgroups to $H$ in $G$, i.e., there are only finitely many subgroups $K$ of $G$ for which there exists an automorphism $\sigma$ of $G$ such that $\sigma(H) = K$.
5 normal subgroup, characteristic core of finite index for every normal subgroup of finite index $H$ in $G$, the characteristic core of $H$ in $G$ has finite index in $G$.
6 normal subgroup contains characteristic subgroup of finite index for every normal subgroup of finite index $H$ in $G$, there exists a characteristic subgroup of finite index $L$ in $G$ such that $L$ is contained in $H$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finite group Group with finitely many homomorphisms to any finite group|FULL LIST, MORE INFO
finitely generated group finitely generated implies every subgroup of finite index has finitely many automorphic subgroups Group with finitely many homomorphisms to any finite group|FULL LIST, MORE INFO
group with finitely many homomorphisms to any finite group finitely many homomorphisms to any finite group implies every subgroup of finite index has finitely many automorphic subgroups every subgroup of finite index has finitely many automorphic subgroups not implies finitely many homomorphisms to any finite group |FULL LIST, MORE INFO
simple group (via finitely many homomorphisms to any finite group)
group of finite composition length (via finitely many homomorphisms to any finite group)