Equivalence of definitions of group with finitely many homomorphisms to any finite group
From Groupprops
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 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 . |