Can someone guide me through this functional equation?

Question

I'm not necessarily sure how to approach this problem—or whether it even has a solution—but I would like to know an example of a non-constant function that satisfies this condition:

$$f(x,y,z)f(x,z,y)f(y,x,z)f(y,z,x)f(z,x,y)f(z,y,x)=1$$

Also, could someone point me to a good book that can help me with these types of questions?

*edit: By the way, I really don't care how this function would look like—if it ends up being a matrix, that'll do.

Thank you all!

**Edit: I probably should clarify that the 1 means identity. I'm putting that info down because I'm not sure what the answer could be, but I would like it to refer as an identity.

Answer 1

If $\,f(x,y,z) := g(x y z)\,$ for some non-constant function $\,g\,$ that satisfies $\,g(x)^6 = 1\,$ then $\,f\,$ satisifes your functional equation. There are other possibilities such as $\, f(x,y,z) := g(x\!+\!y\!+\!z),\,$ or in general, $\,f(x,y,z) := g(h(x,y,z))\,$ where $\,h\,$ is any symmetric function.

Answer 2

How about the non-constant function $$f(x,y,z)=\begin{cases}1&\text{ if } x\leq 0\\-1&\text{ if }x>0\end{cases}.$$