9.3: Ejercicios
- Page ID
- 116020
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1. Dejar\(F\) ser el grupo de todas las funciones desde\([0,1]\) hasta\(\mathbb{R}\text{,}\) debajo de la adición puntual. Let
\ begin {ecuación*} N=\ {f\ in F: f (1/4) =0\}. \ end {ecuación*}
Demostrar que\(F/N\) es un grupo que es isomórfico\(\mathbb{R}\text{.}\)
2. Let\(N=\{1,-1\}\subseteq \mathbb{R}^*\text{.}\) Prove that \(\mathbb{R}^*/N\) is a group that's isomorphic to \(\mathbb{R}^+\text{.}\)
3. Let\(n\in \mathbb{Z}^+\) and let\(H=\{A\in GL(n,\mathbb{R})\,:\, \det A =\pm 1\}\text{.}\) Identificar un grupo familiar para nosotros que es isomórfico para\(GL(n,\mathbb{R})/H\text{.}\)
4. Dejar\(G\) y\(G'\) ser grupos con respectivos subgrupos normales\(N\) y\(N'\text{.}\) Demostrar o desacreditar: Si\(G/N\simeq G'/N'\) entonces\(G\simeq G'\text{.}\)