Group with finitely many homomorphisms to any finite group

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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

Further information: equivalence of definitions of group with finitely many homomorphisms to any finite group

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	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

Facts