Problem 5.4.12 (Automorphism group of a finite cyclic group)

Proof. Let $G\simeq Z_n$ be a cyclic group of order $n$, where the decomposition of $n$ in prime factors is

The Sylow $p$-subgroups $S_i$ of $G$ are cyclic, so $S_i\simeq Z_{p^{{\alpha }_i}}(1\le i\le r)$, and by Exercise 11,

Using Proposition 17 of Section 4, we obtain

• If $n$ is odd, then

  $$\begin{array}{lll}\hfill {(\mathbb{Z}∕\mathit{𝑛\mathbb{Z}})}^{\times }\simeq (\mathrm{Aut}(G)\simeq Z_{p_1^{{\alpha }_1-1}(p_1-1)}\times Z_{p_2^{{\alpha }_2-1}(p_2-1)}\times \cdots \times Z_{p_r^{{\alpha }_r-1}(p_r-1)}.& & \hfill \end{array}$$

• If $n$ is even, then $p_1=2$, and

  • If ${\alpha }_1>2$,

    $${(\mathbb{Z}∕\mathit{𝑛\mathbb{Z}})}^{\times }\simeq \mathrm{Aut}(G)\simeq (Z_2\times Z_{2^{{\alpha }_1-2}})\times Z_{p_2^{{\alpha }_2-1}(p_2-1)}\times \cdots \times Z_{p_r^{{\alpha }_r-1}(p_r-1)}.$$

  • If ${\alpha }_1=2$,

    $${(\mathbb{Z}∕\mathit{𝑛\mathbb{Z}})}^{\times }\simeq \mathrm{Aut}(G)\simeq Z_2\times Z_{p_2^{{\alpha }_2-1}(p_2-1)}\times \cdots \times Z_{p_r^{{\alpha }_r-1}(p_r-1)}.$$

  • If ${\alpha }_1=1$,

    $${(\mathbb{Z}∕\mathit{𝑛\mathbb{Z}})}^{\times }\simeq \mathrm{Aut}(G)\simeq Z_{p_2^{{\alpha }_2}(p_2-1)}\times \cdots \times Z_{p_r^{{\alpha }_r-1}(p_r-1)}.$$