A group is Hopfian when every surjective endomorphism is an automorphism. Every finitely generated residually finite group is Hopfian.
A non-Hopfian group admits a surjective endomorphism which is not injective. The Baumslag-Solitar group has such an endomorphism , : its image contains and , while the nonidentity commutator lies in its kernel by Britton's lemma.
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A Hopfian group is a type of group that satisfies a specific property related to its endomorphisms. Specifically, a group \( G \) is called a Hopfian group if every surjective (onto) endomorphism \( f: G \to G \) is an isomorphism.