Residually finite not implies Hopfian
This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., residually finite group) need not satisfy the second group property (i.e., Hopfian group)
View a complete list of group property non-implications | View a complete list of group property implications
Get more facts about residually finite group|Get more facts about Hopfian group
|residually finite group||every non-identity element is outside of some normal subgroup of finite index, i.e., there is a homomorphism to a finite group where it has a non-identity image.|
|Hopfian group||every surjective endomorphism is an automorphism.|
- This is residually finite: For any non-identity element, find any coordinate where it takes a non-identity value and consider the normal subgroup of those elements that are the identity at that coordinate. (Equivalently, the quotient map here is projection on that coordinate).
- This is not Hopfian: Suppose is the indexing set. Pick and consider a bijection . By coordinate shifting, this induces an isomorphism from the quotient by the coordinate subgroup to the whole group, and hence, a surjective endomorphism of the whole group that is not injective. Explicitly, if are all copies of a nontrivial finite group , and is the product of the s, then a bijection induces an isomorphism which thus gives a surjective endomorphism from to with nontrivial kernel.