15.5: Un proyecto
- Page ID
- 111250
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)El objetivo principal de la teoría de grupos finitos es clasificar todos los grupos finitos posibles hasta el isomorfismo. Este problema es muy difícil aunque intentemos clasificar los grupos de orden menores o iguales a\(60\text{.}\) Sin embargo, podemos descomponer el problema en varios problemas intermedios. Se trata de un proyecto desafiante que requiere un conocimiento práctico de la teoría de grupos que has aprendido hasta este punto. Aunque no lo completes, te enseñará mucho sobre grupos finitos. Puedes usar Mesa\(15.21\) como guía.
\(Table \text { } 15.21\). Número de grupos distintos\(G\text{,}\)\(|G| \leq 60\)
| Orden | Número | Orden | Número | Orden | Número | Orden | Número |
| \(1\) | ? | \(16\) | \(14\) | \(31\) | \(1\) | \(46\) | \(2\) |
| \(2\) | ? | \(17\) | \(1\) | \(32\) | \(51\) | \(47\) | \(1\) |
| \(3\) | ? | \(18\) | ? | \(33\) | \(1\) | \(48\) | \(52\) |
| \(4\) | ? | \(19\) | ? | \(34\) | ? | \(49\) | ? |
| \(5\) | ? | \(20\) | \(5\) | \(35\) | \(1\) | \(50\) | \(5\) |
| \(6\) | ? | \(21\) | ? | \(36\) | \(14\) | \(51\) | ? |
| \(7\) | ? | \(22\) | \(2\) | \(37\) | \(1\) | \(52\) | ? |
| \(8\) | ? | \(23\) | \(1\) | \(38\) | ? | \(53\) | ? |
| \(9\) | ? | \(24\) | ? | \(39\) | \(2\) | \(54\) | \(15\) |
| \(10\) | ? | \(25\) | \(2\) | \(40\) | \(14\) | \(55\) | \(2\) |
| \(11\) | ? | \(26\) | \(2\) | \(41\) | \(1\) | \(56\) | ? |
| \(12\) | \(5\) | \(27\) | \(5\) | \(42\) | ? | \(57\) | \(2\) |
| \(13\) | ? | \(28\) | ? | \(43\) | \(1\) | \(58\) | ? |
| \(14\) | ? | \(29\) | \(1\) | \(44\) | \(4\) | \(59\) | \(1 \) |
| \(15\) | \(1\) | \(30\) | \(4\) | \(45\) | ? | \(60\) | \(13\) |
Encuentra todos los grupos simples\(G\) (\(|G| \leq 60\)). No utilice el Teorema del Orden Odd a menos que esté preparado para demostrarlo.
Encuentra el número de grupos distintos\(G\text{,}\) donde el orden de\(G\) es\(n\) para\(n = 1, \ldots, 60\text{.}\)
Encuentra los grupos reales (hasta isomorfismo) para cada\(n\text{.}\)