Define $T:\mathbb{Z}_{12}\longrightarrow\mathbb{Z}_{12}$. Then find the number of inner automorphisms.

Question

Question Define $T:\mathbb{Z}_{12}\longrightarrow\mathbb{Z}_{12}$. Then find the number of inner automorphisms.

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MY Approach $\mathbb{Z}_{12}$ is cyclic $\Longrightarrow\mathbb{Z}_{12}$ is abelian $\Longrightarrow$ If $G=\mathbb{Z}_{12}$, then $G=Z\left(G\right)$, Center of G.

We know that $G/Z\left(G\right)$ is isomorphic to $Inn\left(G\right)$.

$|Inn\left(G\right)|=|G/Z\left(G\right)|=\frac{|G|}{Z\left(G\right)}=1$.

Book mentions answer is $6$.

Answer 1

Your argument is complicated in a needless way, but it is correct: the only inner automorphism of $\mathbb{Z}_{12}$ is the identity.

Note that, since $\mathbb{Z}_{12}$ is a quotient of $\mathbb Z$, which is abelian, $\mathbb{Z}_{12}$ is abelian. On the other hand, an inner automorphism of a group $G$ is a map from $G$ into itself of the type $g\mapsto hgh^{-1}$. But then when $G$ is abelian this map is the identity, since $hgh^{-1}=hh^{-1}g=g$.