Automorphism Group/Examples/Cyclic Group C3/Proof 2
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Example of Automorphism Group
Consider the cyclic group $C_3$, which can be presented as its Cayley table:
- $\begin {array} {r|rrr} \struct {\Z_3, +_3} & \eqclass 0 3 & \eqclass 1 3 & \eqclass 2 3 \\ \hline \eqclass 0 3 & \eqclass 0 3 & \eqclass 1 3 & \eqclass 2 3 \\ \eqclass 1 3 & \eqclass 1 3 & \eqclass 2 3 & \eqclass 0 0 \\ \eqclass 2 3 & \eqclass 2 3 & \eqclass 0 3 & \eqclass 1 3 \\ \end {array}$
The automorphism group of $C_3$ is given by:
- $\Aut {C_3} = \set {\phi, \theta}$
where $\phi$ and $\theta$ are defined as:
\(\ds \map \phi {\eqclass 0 3}\) | \(=\) | \(\ds \eqclass 0 3\) | ||||||||||||
\(\ds \map \phi {\eqclass 1 3}\) | \(=\) | \(\ds \eqclass 1 3\) | ||||||||||||
\(\ds \map \phi {\eqclass 2 3}\) | \(=\) | \(\ds \eqclass 2 3\) |
\(\ds \map \theta {\eqclass 0 3}\) | \(=\) | \(\ds \eqclass 0 3\) | ||||||||||||
\(\ds \map \theta {\eqclass 1 3}\) | \(=\) | \(\ds \eqclass 2 3\) | ||||||||||||
\(\ds \map \theta {\eqclass 2 3}\) | \(=\) | \(\ds \eqclass 1 3\) |
The Cayley table of $\Aut {C_3}$ is then:
- $\begin{array}{r|rr}
& \phi & \theta \\
\hline \phi & \phi & \theta \\ \theta & \theta & \phi \\ \end{array}$
Proof
This is an example of Order of Automorphism Group of Prime Group:
- $\order {\Aut G} = p - 1$
for a group of prime order $p$.
The only group of order $2$ is the cyclic group of order $2$.
The result follows.
$\blacksquare$