# Group with finitely many homomorphisms to any finite group

## Definition

### Equivalent definitions in tabular format

No. Shorthand A group $G$ is termed a group with finitely many homomorphisms to any finite group if ...
1 finitely many homomorphisms to any finite group for any finite group $K$, there are only finitely many homomorphisms from $G$ to $K$.
2 finitely many surjective homomorphisms to any finite group for any finite group $K$, there are only finitely many surjective homomorphisms from $G$ to $K$.
3 finitely many normal subgroups of fixed finite index for any natural number $n$, $G$ has only finitely many normal subgroups of finite index with the index equal to $n$.
4 finitely many normal subgroups of bounded finite index for any natural number $n$, $G$ has only finitely many normal subgroups of finite index with the index at most $n$.
5 finitely many subgroups of fixed finite index for any natural number $n$, $G$ has only finitely many subgroups of finite index with the index equal to $n$.
6 finitely many subgroups of bounded finite index for any natural number $n$, $G$ has only finitely many subgroups of finite index with the index at most $n$.
7 intersection of normal subgroups of fixed finite index has finite index for any natural number $n$, the intersection of all the normal subgroups of $G$ of index equal to $n$ is also a normal subgroup of finite index in $G$.
8 intersection of subgroups of fixed finite index has finite index for any natural number $n$, the intersection of all the subgroups of $G$ of index equal to $n$ is also a subgroup of finite index in $G$.
9 intersection of normal subgroups of bounded finite index has finite index for any natural number $n$, the intersection of all the normal subgroups of $G$ of index at most $n$ is also a normal subgroup of finite index in $G$.
10 intersection of subgroups of bounded finite index has finite index for any natural number $n$, the intersection of all the subgroups of $G$ of index at most $n$ is also a subgroup of finite index in $G$.

### Equivalence of definitions

The nontrivial ingredients in the equivalences are Poincare's theorem (which asserts that a subgroup of finite index $n$ contains a normal subgroup of finite index at most $n!$) and that index satisfies intersection inequality, which yields that the intersection of finitely many subgroups of finite index again has finite index. We also need to use the fact that a subgroup of finite index can be contained in only finitely many intermediate subgroups.