# Equivalence of definitions of group with finitely many homomorphisms to any finite group

This article gives a proof/explanation of the equivalence of multiple definitions for the term group with finitely many homomorphisms to any finite group
View a complete list of pages giving proofs of equivalence of definitions

## The definitions that we have to prove as equivalent

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 $G$.