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

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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.