Olympiad problems

1986 Traian Lălescu, 1.1

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.

2018 Ramnicean Hope, 3

Tags: automorphism , morphism , ring theory , endomorphism , polynomial ring , function

[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]

2017 District Olympiad, 2

Tags: superior algebra , morphism , group theory , abstract algebra

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.

2008 Gheorghe Vranceanu, 2

Tags: superior algebra , group theory , morphism , transposition decomposition , algebra

Prove that the only morphisms from a finite symmetric group to the multiplicative group of rational numbers are the identity and the signature.

2000 Romania National Olympiad, 3

Tags: abstract algebra , homomorphism , superior algebra , morphism , group theory

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]

2004 Nicolae Coculescu, 3

Tags: function , group theory , morphism , endomorphism

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.

2011 District Olympiad, 2

Tags: group theory , morphism , superior algebra

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\} $

2002 District Olympiad, 2

Tags: superior algebra , product rings , ring theory , morphism

[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.