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

## 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 $H$.