Groupprops, The Group Properties Wiki (pre-alpha)

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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)
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Contents 1 Statement 2 Related facts 3 Facts used 4 Proof

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).