# Residually finite not implies Hopfian

From Groupprops

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) neednotsatisfy the second group property (i.e., Hopfian group)

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## Statement

A residually finite group need not be a Hopfian group.

## Definitions used

Term | Definition used |
---|---|

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. |

## Related facts

## Proof

Consider an infinite-dimensional vector space over a field, or more generally, any infinite external direct power or restricted external direct power of a nontrivial finite group

- 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.