11.5: Ejercicios Adicionales- Automorfismos
- Page ID
- 111066
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)\(\operatorname{Aut}(G)\)Sea el conjunto de todos los automorfismos de\(G\text{;}\) eso es, isomorfismos de\(G\) a sí mismo. Demostrar que este conjunto forma un grupo y es un subgrupo del grupo de permutaciones de es\(G\text{;}\) decir,\(\operatorname{Aut}(G) \leq S_G\text{.}\)
Un automorfismo interno de\(G\text{,}\)
\[ i_g : G \rightarrow G\text{,} \nonumber \]
está definido por el mapa
\[ i_g(x) = g x g^{-1}\text{,} \nonumber \]
para\(g \in G\text{.}\) Demostrar que\(i_g \in \operatorname{Aut}(G)\text{.}\)
El conjunto de todos los automorfismos internos se denota por\(\operatorname{Inn}(G)\text{.}\) Show que\(\operatorname{Inn}(G)\) es un subgrupo de\(\operatorname{Aut}(G)\text{.}\)
Encuentra un automorfismo de un grupo\(G\) que no sea un automorfismo interno.
Dejar\(G\) ser un grupo y\(i_g\) ser un automorfismo interno de\(G\text{,}\) y definir un mapa
\[ G \rightarrow \operatorname{Aut}(G) \nonumber \]
por
\[ g \mapsto i_g\text{.} \nonumber \]
Demostrar que este mapa es un homomorfismo con imagen\(\operatorname{Inn}(G)\) y kernel\(Z(G)\text{.}\) Usa este resultado para concluir que
\[ G/Z(G) \cong \operatorname{Inn}(G)\text{.} \nonumber \]
Compute\(\operatorname{Aut}(S_3)\) y\(\operatorname{Inn}(S_3)\text{.}\) haga lo mismo para\(D_4\text{.}\)
Encuentra todos los homomorfismos\(\phi : {\mathbb Z} \rightarrow {\mathbb Z}\text{.}\) Qué es\(\operatorname{Aut}({\mathbb Z})\text{?}\)
Encuentra todos los automorfismos de\({\mathbb Z}_8\text{.}\) Probarlo\(\operatorname{Aut}({\mathbb Z}_8) \cong U(8)\text{.}\)
Para\(k \in {\mathbb Z}_n\text{,}\) definir un mapa\(\phi_k : {\mathbb Z}_n \rightarrow {\mathbb Z}_n\) por\(a \mapsto ka\text{.}\) Demostrar que\(\phi_k\) es un homomorfismo.
Demostrar que\(\phi_k\) es un isomorfismo si y solo si\(k\) es un generador de\({\mathbb Z}_n\text{.}\)
Demostrar que cada automorfismo de\({\mathbb Z}_n\) es de la forma\(\phi_k\text{,}\) donde\(k\) es un generador de\({\mathbb Z}_n\text{.}\)
Demostrar que\(\psi : U(n) \rightarrow \operatorname{Aut}({\mathbb Z}_n)\) es un isomorfismo, donde\(\psi : k \mapsto \phi_k\text{.}\)