Olympiad problems

Found problems: 2

1985 Traian Lălescu, 2.3

Let $ 0\neq\varrho\in\text{Hom}\left( \mathbb{Z}_4,\mathbb{Z}_2\right) ,$ $ \text{id}\neq\iota\in\text{Aut}\left( \mathbb{Z}_4\right) ,$ $ G:=\left\{ (x,y)\in\mathbb{Z}_4^2\big|x-y\in\ker\varrho\right\} , $ and $ \rho_1,\rho_2, $ the canonic projections of $ G $ into $ \mathbb{Z}_4. $ Prove that there exists an unique $ \nu\in\text{Hom}\left( \mathbb{Z}_4,G\right) $ such that $ \rho_1\circ\nu=\text{id} $ and $ \rho_2\circ\nu =\iota . $ Determine numerically this morphism.

2000 Romania National Olympiad, 3

We say that the abelian group $ G $ has property [i](P)[/i] 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: [b]a)[/b] the group $ \left( \mathbb{Q}^*,\cdot \right) $ hasn’t property [i](P).[/i] [b]b)[/b] the group $ \left( \mathbb{Q}, +\right) $ has property [i](P).[/i]

group theory abstract algebra morphism homomorphism superior algebra