11.5: Additional Exercises- Automorphisms
- Page ID
- 81120
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Let \(\operatorname{Aut}(G)\) be the set of all automorphisms of \(G\text{;}\) that is, isomorphisms from \(G\) to itself. Prove this set forms a group and is a subgroup of the group of permutations of \(G\text{;}\) that is, \(\operatorname{Aut}(G) \leq S_G\text{.}\)
An inner automorphism of \(G\text{,}\)
\[ i_g : G \rightarrow G\text{,} \nonumber \]
is defined by the map
\[ i_g(x) = g x g^{-1}\text{,} \nonumber \]
for \(g \in G\text{.}\) Show that \(i_g \in \operatorname{Aut}(G)\text{.}\)
The set of all inner automorphisms is denoted by \(\operatorname{Inn}(G)\text{.}\) Show that \(\operatorname{Inn}(G)\) is a subgroup of \(\operatorname{Aut}(G)\text{.}\)
Find an automorphism of a group \(G\) that is not an inner automorphism.
Let \(G\) be a group and \(i_g\) be an inner automorphism of \(G\text{,}\) and define a map
\[ G \rightarrow \operatorname{Aut}(G) \nonumber \]
by
\[ g \mapsto i_g\text{.} \nonumber \]
Prove that this map is a homomorphism with image \(\operatorname{Inn}(G)\) and kernel \(Z(G)\text{.}\) Use this result to conclude that
\[ G/Z(G) \cong \operatorname{Inn}(G)\text{.} \nonumber \]
Compute \(\operatorname{Aut}(S_3)\) and \(\operatorname{Inn}(S_3)\text{.}\) Do the same thing for \(D_4\text{.}\)
Find all of the homomorphisms \(\phi : {\mathbb Z} \rightarrow {\mathbb Z}\text{.}\) What is \(\operatorname{Aut}({\mathbb Z})\text{?}\)
Find all of the automorphisms of \({\mathbb Z}_8\text{.}\) Prove that \(\operatorname{Aut}({\mathbb Z}_8) \cong U(8)\text{.}\)
For \(k \in {\mathbb Z}_n\text{,}\) define a map \(\phi_k : {\mathbb Z}_n \rightarrow {\mathbb Z}_n\) by \(a \mapsto ka\text{.}\) Prove that \(\phi_k\) is a homomorphism.
Prove that \(\phi_k\) is an isomorphism if and only if \(k\) is a generator of \({\mathbb Z}_n\text{.}\)
Show that every automorphism of \({\mathbb Z}_n\) is of the form \(\phi_k\text{,}\) where \(k\) is a generator of \({\mathbb Z}_n\text{.}\)
Prove that \(\psi : U(n) \rightarrow \operatorname{Aut}({\mathbb Z}_n)\) is an isomorphism, where \(\psi : k \mapsto \phi_k\text{.}\)