# automorphism proof

• Oct 31st 2010, 10:01 PM
Danneedshelp
automorphism proof
Theorem: for any positive integer n, $Aut(Z_{n})$ is isomorphic to $U(n)$.

Q: Prove that every element $g\in\\Aut(Z_{n})$ is determined by $g(1)$; meaning, $g(1)$ results in a well-defined automorphism of $Z_{n}$.

A: let $g\in\\Aut(Z_{n})$ be arbitrary. Then $f:Z_{n}\rightarrow\Z_{n}$ is an isomorphism. So, to show if something is well-defined I need to show if a=c, b=d, then a*b=c*d. I am not sure what to my a,b, and c's are. I know $f(k)=kf(1)$. so, do I let $f(m)=mf(1)$ and $f(p)=pf(1)$. Now, I want to show $f(p)*f(q)=mf(1)*pf(1)$?

Thanks
• Oct 31st 2010, 10:07 PM
Drexel28
Quote:

Originally Posted by Danneedshelp
Theorem: for any positive integer n, $Aut(Z_{n})$ is isomorphic to $U(n)$.

Q: Prove that every element $g\in\\Aut(Z_{n})$ is determined by $g(1)$; meaning, $g(1)$ results in a well-defined automorphism of $Z_{n}$.

A: let $g\in\\Aut(Z_{n})$ be arbitrary. Then $f:Z_{n}\rightarrow\Z_{n}$ is an isomorphism. So, to show if something is well-defined I need to show if a=c, b=d, then a*b=c*d. I am not sure what to my a,b, and c's are. I know $f(k)=kf(1)$. so, do I let $f(m)=mf(1)$ and $f(p)=pf(1)$. Now, I want to show $f(p)*f(q)=mf(1)*pf(1)$?

Thanks

I'm sorry, what is $U(n)$? That can't be universal notation considering the very common Lie group called the Unitary Group is denoted $U(n)$.
• Oct 31st 2010, 10:22 PM
Danneedshelp
$U(n)$ represents the group of units modulo n. So, for example, $U(10)=\{1,3,7,9\}$.

Sorry for not stating that.
• Nov 1st 2010, 08:23 AM
roninpro
I think that your analysis of the problem is a little bit too complicated. For an easy example, let us look at the group $\mathbb{Z}_{10}$. If $f:\mathbb{Z}_{10}\to \mathbb{Z}_{10}$ is an automorphism, it must send a generator to a generator. To be explicit, a number $a$ is a generator of $\mathbb{Z}_n$ if and only if $\gcd(a,n)=1$. In our case, $a$ is a generator if and only if $\gcd(a,10)=1$. This means that $a=1, 3, 5, 7, 9$. So, we have the following possibilities: $f(1)=1$, $f(1)=3$, $f(1)=5$, $f(1)=7$, or $f(1)=9$. If you try composing these maps, you will find that you recover the multiplication table for your $U_{10}$.

Give it a try. Good luck!
• Nov 1st 2010, 08:28 AM
Swlabr
Quote:

Originally Posted by Drexel28
I'm sorry, what is $U(n)$? That can't be universal notation considering the very common Lie group called the Unitary Group is denoted $U(n)$.

$U(R)$ is common for denoting the group of units of a ring. I suppose this is just a perversion of that.