Problem

Source: Romanian National Olympiad 2000, Grade XII, Problem 3

Tags: abstract algebra, group theory, superior algebra, morphisms, homomorphisms



We say that the abelian group $ G $ has property (P) if, for any commutative group $ H, $ any $ H’\le H $ and any momorphism $ \mu’:H\longrightarrow G, $ there exists a morphism $ \mu :H\longrightarrow G $ such that $ \mu\bigg|_{H’} =\mu’ . $ Show that: a) the group $ \left( \mathbb{Q}^*,\cdot \right) $ hasn’t property (P). b) the group $ \left( \mathbb{Q}, +\right) $ has property (P).