# Group with finitely many homomorphisms to any finite group

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 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.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finite group
finitely generated group finitely generated implies finitely many homomorphisms to any finite group finitely many homomorphisms to any finite group not implies finitely generated |FULL LIST, MORE INFO
simple group simple implies finitely many homomorphisms to any finite group (obvious) |FULL LIST, MORE INFO
group of finite composition length finite composition length implies finitely many homomorphisms to any finite group

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group in which every subgroup of finite index has finitely many automorphic subgroups finitely many homomorphisms to any finite group implies every subgroup of finite index has finitely many automorphic subgroups