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3.4: Ejercicios

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    116039
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    1. Verdadero/Falso. Para cada una de las siguientes, escribe T si la afirmación es verdadera; de lo contrario, escribe F. NO es necesario dar explicaciones ni mostrar trabajo para este problema. A lo largo, dejar\(G\) y\(G'\) ser grupos.
    1. Si existe un homomorfismo\(\phi\,:\,G\to G'\text{,}\) entonces\(G\) y\(G'\) deben ser grupos isomórficos.
    2. Hay un entero\(n\geq 2\) tal que\(\mathbb{Z}\simeq \mathbb{Z}_n\text{.}\)
    3. Si\(|G|=|G'|=3\text{,}\) entonces debemos tener\(G\simeq G'\text{.}\)
    4. Si\(|G|=|G'|=4\text{,}\) entonces debemos tener\(G\simeq G'\text{.}\)

     

    1. Para cada una de las siguientes funciones, probar o desmentir que la función es (i) un homomorfismo; (ii) un isomorfismo. (¡Recuerda trabajar con la operación predeterminada en cada uno de estos grupos!)
    1. La función\(f:\mathbb{Z}\to\mathbb{Z}\) definida por\(f(n)=2n\text{.}\)
    2. La función\(g:\mathbb{R}\to\mathbb{R}\) definida por\(g(x)=x^2\text{.}\)
    3. La función\(h:\mathbb{Q}^*\to\mathbb{Q}^*\) definida por\(h(x)=x^2\text{.}\)

     

    1. Definir\(d : GL(2,\mathbb{R})\to \mathbb{R}^*\) por\(d(A)=\det A\text{.}\) probar/desacreditar que\(d\) es:
    1. un homomorfismo
    2. 1-1
    3. onto
    4. un isomorfismo.
    1. Completa las tablas de grupo para\(\mathbb{Z}_4\) y\(\mathbb{Z}_8^{\times}\text{.}\) Usa las tablas de grupo para decidir si\(\mathbb{Z}_8^{\times}\) son isomórficas entre sí o no\(\mathbb{Z}_4\). (No es necesario que proporcione una prueba.)
    1. Vamos a\(n\in \mathbb{Z}^+\text{.}\) Demostrar que\(\langle n\mathbb{Z},+\rangle \simeq \langle \mathbb{Z},+\rangle\text{.}\)
    2.  
    1. Dejar\(G\) y\(G'\) ser grupos, donde\(G\) es abeliano y\(G\simeq G'\text{.}\) Demostrar que\(G'\) es abeliano.
    2. Dar un ejemplo de grupos\(G\) y\(G'\text{,}\) donde\(G\) es abeliano y existe un homomorfismo desde\(G\) hasta\(G'\text{,}\) pero NO\(G'\) es abeliano.
    1. Dejar\(\langle G,\cdot\rangle\) y\(\langle G',\cdot'\rangle\) ser grupos con elementos de identidad\(e\) y\(e'\text{,}\) respectivamente, y dejar\(\phi\) ser un homomorfismo de\(G\) a\(G'\text{.}\) Let\(a\in G\text{.}\) Prove that\(\phi(a)^{-1}=\phi(a^{-1})\text{.}\)

    This page titled 3.4: Ejercicios is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Jessica K. Sklar via source content that was edited to the style and standards of the LibreTexts platform.