Found problems: 8
[b]a)[/b] Let $ u $ be a polynom in $ \mathbb{Q}[X] . $ Prove that the function $ E_u:\mathbb{Q}[X]\longrightarrow\mathbb{Q}[X] $ defined as $ E_u(P)=P(u) $ is an endomorphism.
[b]b)[/b] Let $ E $ be an injective endomorphism of $ \mathbb{Q} [X] . $ Show that there exists a nonconstant polynom $ v $ in $ \mathbb{Q}[X] $ such that $ E(P)=P(v) , $ for any $ P $ in $ \mathbb{Q}[X] . $
[b]c)[/b] Let $ A $ be an automorphism of $ \mathbb{Q}[X] . $ Demonstrate that there is a nonzero constant polynom $ w $ in $ \mathbb{Q}[X] $ which has the property that $ A(P)=P(w) , $ for any $ P $ in $ \mathbb{Q}[X] . $
[i]Marcel Țena[/i]
Let be a finite group $ G $ having an endomorphism $ \eta $ that has exactly one fixed point.
[b]a)[/b] Demonstrate that the function $ f:G\longrightarrow G $ defined as $ f(x)=x^{-1}\cdot\eta (x) $ is bijective.
[b]b)[/b] Show that $ G $ is commutative if the composition of the function $ f $ from [b]a)[/b] with itself is the identity function.
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]
[b]a)[/b] Show that, for any distinct natural numbers $ m,n, $ the rings $ \mathbb{Z}_2\times \underbrace{\cdots}_{m\text{ times}} \times\mathbb{Z}_2,\mathbb{Z}_2\times \underbrace{\cdots}_{n\text{ times}} \times\mathbb{Z}_2 $ are homomorphic, but not isomorphic.
[b]b)[/b] Show that there are infinitely many pairwise nonhomomorphic rings of same order.
Let be two nontrivial rings linked by an application ($ K\stackrel{\vartheta }{\mapsto } L $) having the following properties:
$ \text{(i)}\quad x,y\in K\implies \vartheta (x+y) = \vartheta (x) +\vartheta (y) $
$ \text{(ii)}\quad \vartheta (1)=1 $
$ \text{(iii)}\quad \vartheta \left( x^3\right) =\vartheta^3 (x) $
[b]a)[/b] Show that if $ \text{char} (L)\ge 4, $ and $ K,L $ are fields, then $ \vartheta $ is an homomorphism.
[b]b)[/b] Prove that if $ K $ is a noncommutative division ring, then it’s possible that $ \vartheta $ is not an homomorphism.
Prove that the only morphisms from a finite symmetric group to the multiplicative group of rational numbers are the identity and the signature.
Let $ G $ be the set of matrices of the form $ \begin{pmatrix} a&b\\0&1 \end{pmatrix} , $ with $ a,b\in\mathbb{Z}_7,a\neq 0. $
[b]a)[/b] Verify that $ G $ is a group.
[b]b)[/b] Show that $ \text{Hom}\left( (G,\cdot) ; \left( \mathbb{Z}_7,+ \right) \right) =\{ 0\} $
Let be a group and two coprime natural numbers $ m,n. $ Show that if the applications $ G\ni x\mapsto x^{m+1},x^{n+1} $ are surjective endomorphisms, then the group is commutative.