Definition
Equivalent definitions in tabular format
No. 
Shorthand 
A group 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 , there are only finitely many homomorphisms from to .

2 
finitely many surjective homomorphisms to any finite group 
for any finite group , there are only finitely many surjective homomorphisms from to .

3 
finitely many normal subgroups of fixed finite index 
for any natural number , has only finitely many normal subgroups of finite index with the index equal to .

4 
finitely many normal subgroups of bounded finite index 
for any natural number , has only finitely many normal subgroups of finite index with the index at most .

5 
finitely many subgroups of fixed finite index 
for any natural number , has only finitely many subgroups of finite index with the index equal to .

6 
finitely many subgroups of bounded finite index 
for any natural number , has only finitely many subgroups of finite index with the index at most .

7 
intersection of normal subgroups of fixed finite index has finite index 
for any natural number , the intersection of all the normal subgroups of of index equal to is also a normal subgroup of finite index in .

8 
intersection of subgroups of fixed finite index has finite index 
for any natural number , the intersection of all the subgroups of of index equal to is also a subgroup of finite index in .

9 
intersection of normal subgroups of bounded finite index has finite index 
for any natural number , the intersection of all the normal subgroups of of index at most is also a normal subgroup of finite index in .

10 
intersection of subgroups of bounded finite index has finite index 
for any natural number , the intersection of all the subgroups of of index at most is also a subgroup of finite index in .

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 contains a normal subgroup of finite index at most ) 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
Weaker properties
Facts