Let $f =
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 &7 & 8 & 9 \\
7 & 9 & 5 & 8 & 3 & 6 & 1 & 4 & 2
\end{pmatrix}
$ and $g =
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 &7 & 8 & 9 \\
9 & 8 & 5 & 1 & 2 & 4 & 7 & 3 & 6
\end{pmatrix}$.
How do I find elements of the cyclic group $\langle f\rangle$ generated by $f$? Similarly, how do I find elements of the cyclic group $\langle g\rangle$ generated by $g$?
First I wrote $f$ and $g$ as a product of disjoint cycles: $f=(17)(29)(35)(48)$ and $g=(1964)(2835)$. Then I can see that their orders are $2$ and $4$ respectively. Now I am stuck. Would appreciate any help.
Best Answer
As $f$ is a product of disjoint transpositions the cyclic group generated by $f$ is equal to the set containing the identity and $f$ itself.
The cyclic group generated by $g$ is the group $\{Id, g, g^2, g^3\}$ as $g$ is the product of two disjoint $4$-cycles.