6.6: Ejercicios
- Page ID
- 115988
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- Vamos\(\sigma=(134)\text{,}\)\(\tau=(23)(145)\text{,}\)\(\rho=(56)(78)\text{,}\) y\(\alpha=(12)(145)\) en\(S_8\text{.}\) Compute lo siguiente.
- \(\sigma \tau\)
- \(\tau \sigma\)
- \(\tau^2\)
- \(\tau^{-1}\)
- \(o(\tau)\)
- \(o(\rho)\)
- \(o(\alpha)\)
- \(\langle \tau\rangle\)
2. Demostrar Lema 6.3.4.
3. Demostrar que\(A_n\) es un subgrupo de\(S_n\text{.}\)
4. Demostrar o desacreditar: El conjunto de todas las permutaciones impares en\(S_n\) is a subgroup of \(S_n\text{.}\)
5. Dejar\(n\) ser un entero mayor que 2. \(m \in \{1,2,\ldots,n\}\text{,}\)y let\(H=\{\sigma\in S_n\,:\,\sigma(m)=m\}\) (en otras palabras,\(H\) es el conjunto de todas las permutaciones en\(S_n\) ese arreglo\(m\)).
- Demostrar que\(H\leq S_n\text{.}\)
- Identificar un grupo familiar al que\(H\) sea isomórfico. (No es necesario mostrar ningún trabajo.)
6. Escribir\(rfr^2frfr\)\(D_5\) en forma estándar.
7. Demostrar o desacreditar:\(D_6\simeq S_6\text{.}\)
8. Qué elementos de\(D_4\) (si los hay)
- tener orden\(2\)?
- tener orden\(3\text{?}\)
9. \(n\)Sea un entero par que sea mayor o igual a\(4\). Demostrar\(r^{n/2}\in Z(D_n)\text{:}\) que es decir, demostrar\(r^{n/2}\) que se desplaza con cada elemento de\(D_n\text{.}\) (NO se limite a referirse a la última declaración en Teorema\(6.5.4\); esa es la afirmación que está demostrando aquí.)