automorphism group of a cyclic group
Theorem 1.
The automorphism group^{} of the cyclic group^{} $\mathrm{Z}\mathrm{/}n\mathit{}\mathrm{Z}$ is ${\mathrm{(}\mathrm{Z}\mathrm{/}n\mathit{}\mathrm{Z}\mathrm{)}}^{\mathrm{\times}}$, which is of order $\varphi \mathit{}\mathrm{(}n\mathrm{)}$ (here $\varphi $ is the Euler totient function).
Proof.
Choose a generator^{} $x$ for $\mathbb{Z}/n\mathbb{Z}$. If $\rho \in \mathrm{Aut}(\mathbb{Z}/n\mathbb{Z})$, then $\rho (x)={x}^{a}$ for some integer $a$ (defined up to multiples^{} of $n$); further, since $x$ generates $\mathbb{Z}/n\mathbb{Z}$, it is clear that $a$ uniquely determines $\rho $. Write ${\rho}_{a}$ for this automorphism^{}. Since ${\rho}_{a}$ is an automorphism, ${x}^{a}$ is also a generator, and thus $a$ and $n$ are relatively prime^{1}^{1} If they were not, say $(a,n)=d$, then ${({x}^{a})}^{n/d}={({x}^{a/d})}^{n}=1$ so that ${x}^{a}$ would not generate.. Clearly, then, every $a$ relatively prime to $n$ induces an automorphism. We can therefore define a surjective map
$$\mathrm{\Phi}:\mathrm{Aut}(\mathbb{Z}/n\mathbb{Z})\to {(\mathbb{Z}/n\mathbb{Z})}^{\times}:{\rho}_{a}\mapsto a\phantom{\rule{veryverythickmathspace}{0ex}}(modn)$$ |
$\mathrm{\Phi}$ is also obviously injective, so all that remains is to show that it is a group homomorphism. But for every $a,b\in {(\mathbb{Z}/n\mathbb{Z})}^{\times}$, we have
$$({\rho}_{a}\circ {\rho}_{b})(x)={\rho}_{a}({x}^{b})={({x}^{b})}^{a}={x}^{ab}={\rho}_{ab}(x)$$ |
and thus
$$\mathrm{\Phi}({\rho}_{a}\circ {\rho}_{b})=\mathrm{\Phi}({\rho}_{ab})=ab\phantom{\rule{veryverythickmathspace}{0ex}}(modn)=\mathrm{\Phi}({\rho}_{a})\mathrm{\Phi}({\rho}_{b})$$ |
∎
References
- 1 Dummit, D., Foote, R.M., Abstract Algebra, Third Edition, Wiley, 2004.
Title | automorphism group of a cyclic group |
---|---|
Canonical name | AutomorphismGroupOfACyclicGroup |
Date of creation | 2013-03-22 18:42:35 |
Last modified on | 2013-03-22 18:42:35 |
Owner | rm50 (10146) |
Last modified by | rm50 (10146) |
Numerical id | 6 |
Author | rm50 (10146) |
Entry type | Theorem |
Classification | msc 20A05 |
Classification | msc 20F28 |