Given a relation on a permutation group $S_n$, I'm interested in solving for an unknown permutation. For a concrete example, let's say $\sigma_1=(12)\in S_3$ (in cycle notation), and the relation is
$$\sigma_1^2 \sigma_2=\sigma_2\sigma_1$$
How can I find a $\sigma_2$ that solves this? (Note that I might be able to do this particular example by inspection, but I'm interested in a general method.)
The only idea I have is to represent the permutations by matrices. So,
$$\rho((12))=\left(\begin{array}{ccc}0&1&0\\1&0&0\\0&0&1 \end{array}\right),$$
which amounts to a linear solution of 6 equations (since the representation should 3x3 symmetric matrices) to find $\sigma_1$, but this doesn't always work, since I don't think all matrices can be written as permutations matrices. For example, I think I determined the solution here is
$$\rho(\sigma_1)=\left(\begin{array}{ccc}1&1&1\\1&1&1\\1&1&1 \end{array}\right),$$
But I'm not sure that I can actually write that as a product of cycles. Does anyone have a systematic way to solve such relations, or a way to write such representations as cycles? Or am I barking up on of those "uncountable word problem" trees?
You have
$$e=\sigma_1^2 =\sigma_2 \sigma_1 \sigma_2^{-1}$$ where $e$ is the identity permutation while $$e=\sigma_2 \sigma_1 \sigma_2^{-1}= (\sigma_2(1) \ \sigma_2(2))$$ and this equation is impossible as it would mean that $\sigma_2(1)=\sigma_2(2)$ and that $\sigma_2$ won’t be one-to-one. Hence the equation in $\sigma_2$ has no solution.
Not sure that you can have a general method to solve such equations… despite that you could just in the theorical way use brut force and test all $\sigma_2$