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When does an injective group homomorphism have an inverse?

Given two groups $A$ and $B$ and an injective homomorphism $f : A \to B$. When does a homomorphism $g : B \to A$ exist with $g\circ f = \mathrm{id}_A$ (but not necessarily $f\circ g = \mathrm{id}_B$)?

Hans-Peter Stricker
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