11.5: Additional Exercises- Automorphisms
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 \(\psi : k \mapsto \phi_k\text{.}\)