Finitely generated and free implies Hopfian

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finitely generated free group) must also satisfy the second group property (i.e., Hopfian group)
View all group property implications | View all group property non-implications
Get more facts about finitely generated free group|Get more facts about Hopfian group

Statement

Any finitely generated free group is Hopfian: it is not isomorphic to any proper quotient of itself.

Related facts

• Retract of free group is free on fewer generators
• Free quotient group admits a section

Facts used

1. Free implies residually finite
2. Finitely generated and residually finite implies Hopfian

Proof

The proof follows directly by combining facts (1) and (2).