I have four 4-cycles, given by: $(1452),(2563),(4785),(5896)$. I know that the group generated by these guys are $S(9)$ by asking mathematica for the order of the permutation group generated by these four 4-cycles, which came out to be 9!
I am looking for an elegant way to show this statement, but I can't come up with anything. We tried to show directly that we can get a 2-cycle and a 9-cycle without success.
The motivation for the problem is as follows:
9 squares are arranged in a 3 by 3 grid. I will refer to this grid as the "big square".
You have some kind of a picture drawn in the big square.
The individual squares are scrambled in some weird manner.
Is it possible to get the original picture back, using only the operation given by rotating four squares with the the center of the rotation on the vertex of the central square?
So basically:
\begin{pmatrix} 1 & 2 & 3 \\\ 4 & 5 & 6 \\\ 7 & 8 & 9 \end{pmatrix}
can become
\begin{pmatrix} 2 & 5 & 3 \\\ 1 & 4 & 6 \\\ 7 & 8 & 9 \end{pmatrix}
which corresponds to the cycle (1452).