Inner and outer automorphisms

#group_theory

Let $C$ be a (possibly infinite) cyclic group, and let $Aut (C)$ and $Inn (C)$ be the groups of automorphisms and inner automorphisms, respectively. (Recall an automorphism $γ$ is inner if it is given by conjugation: $γ (b) = a b a^{- 1}$ for some $a \in C$ .)

Describe $Aut (C)$ and $Inn (C)$ in familiar terms, as groups you would study in a first algebra course. Prove your result. (Hint: Where do generators go?)
Write $Aut (Z_{12})$ down explicitly, giving its generic name and computing the order of every element. Show all work.

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Let $C$ be a (possibly infinite) cyclic group, and let $\operatorname{Aut}(C)$ and $\operatorname{Inn}(C)$ be the groups of automorphisms and inner automorphisms, respectively. (Recall an automorphism $\gamma$ is {\bfseries inner} if it is given by conjugation: $\gamma(b)=aba^{-1}$ for some $a\in C$.)
\begin{enumerate}[label=\alph*)]
	\item Describe $\operatorname{Aut}(C)$ and $\operatorname{Inn(C)}$ in familiar terms, as groups you would study in a first algebra course. Prove your result. ({\itshape Hint:} Where do generators go?)
	\item Write $\operatorname{Aut}({\bf Z}_{12})$ down explicitly, giving its generic name and computing the order of every element. Show all work.
\end{enumerate}