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# 4.3: Ejercicios

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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. Cada grupo contiene al menos dos subgrupos distintos.
2. Si$$H$$ es un subgrupo propio de grupo$$G$$ y$$G$$ es finito, entonces debemos tener$$|H|\lt |G|\text{.}$$
3. $$7\mathbb{Z}$$es un subgrupo de$$14\mathbb{Z}\text{.}$$
4. Un grupo$$G$$ puede tener dos subgrupos propios distintos que son isomórficos (entre sí).

2. Dar ejemplos específicos y precisos de los siguientes grupos$$G$$ con subgrupos$$H\text{:}$$

1. Un grupo$$G$$ con un subgrupo adecuado$$H$$ de$$G$$ tal manera que$$|H|=|G|\text{.}$$
2. Un grupo$$G$$ de orden$$12$$ que contiene un subgrupo$$H$$ con$$|H|=3\text{.}$$
3. Un grupo no abeliano$$G$$ que contiene un subgrupo abeliano no trivial$$H\text{.}$$
4. Un subgrupo finito$$H$$ de un grupo infinito$$G\text{.}$$

3. Let$$n\in \mathbb{Z}^+\text{.}$$

1. Demostrar que$$n\mathbb{Z} \leq \mathbb{Z}\text{.}$$
2. Demostrar que el conjunto$$H=\{A\in \mathbb{M}_n(\mathbb{R})\,:\,\det A=\pm 1\}$$ es un subgrupo de$$GL(n,\mathbb{R})\text{.}$$

(Nota: ¡Tus pruebas no necesitan ser largas para ser correctas!)

4. Dejar$$n\in \mathbb{Z}^+\text{.}$$ Para cada grupo$$G$$ y subconjunto$$H\text{,}$$ decidir si$$H$$ es o no un subgrupo de$$G\text{.}$$ En los casos en los que no$$H$$ es un subgrupo de$$G\text{,}$$ proporcionar una prueba. (Nota. ¡Tus pruebas no necesitan ser largas para ser correctas!)

1. $$G=\mathbb{R}\text{,}$$$$H=\mathbb{Z}$$
2. $$G=\mathbb{Z}_{15}\text{,}$$$$H=\{0,5,10\}$$
3. $$G=\mathbb{Z}_{15}\text{,}$$$$H=\{0,4,8,12\}$$
4. $$G=\mathbb{C}\text{,}$$$$H=\mathbb{R}^*$$
5. $$G=\mathbb{C}^*\text{,}$$$$H=\{1,i,-1,-i\}$$
6. $$G=\mathbb{M}_n(\mathbb{R})\text{,}$$$$H=GL(n,\mathbb{R})$$
7. $$G=GL(n,\mathbb{R})\text{,}$$$$H=\{A\in \mathbb{M}_n(\mathbb{R})\,:\,\det A = -1\}$$

5. Let$$G$$ and $$G'$$ be groups, let $$\phi$$ be a homomorphism from $$G$$ to $$G'\text{,}$$ and let $$H$$ be a subgroup of $$G\text{.}$$ Prove that $$\phi(H)$$ is a subgroup of $$G'\text{.}$$

6. Dejemos$$G$$ ser un grupo abeliano, y vamos a$$U=\{g\in G\,:\, g^{-1}=g\}.$$ demostrar que$$U$$ es un subgrupo de$$G\text{.}$$

This page titled 4.3: 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.