Experts in algebra please help - Part II after Part I: we would like to know the number of solutions for this set of six of modular N algebraic equations:
$$ x_1 y_2 = x_2 y_1 \pmod N \qquad (1) \\ x_1 y_3=x_3 y_1 \pmod N \qquad (2) \\ x_4 y_1 \neq x_1 y_4 \pmod N \qquad (3) \\ x_2 y_3=x_3 y_2 \pmod N \qquad (4) \\ x_2 y_4 \neq x_4 y_2 \pmod N \qquad (5) \\ x_3 y_4 \neq x_4 y_3 \pmod N \qquad (6) $$
suppose that all variables are module $N$, i.e. $x_j=x_j \pmod N$ and $y_j=y_j \pmod N$. Here there are $x_1,x_2,x_3,x_4, y_1,y_2,y_3,y_4$ eight variables. NOTE: Here Eq(3),(5),(6) are required to be inequality.
Questions:
(a) For $N=2$, how many independent solutions there are?
(b) For $N=3$, how many independent solutions there are?
(c) For a generic prime number $N$, how many independent solutions there are?
I emphasize these questions about counting number(how many) of independent solutions, so explicit solutions for these equations are NOT required.
Hint (what I have known): is discussed in Part I, an easier question. Here this one is less clear to me. I only know this number $\leq N^8$.
So what is the exact number expression of this number of independent solutions? Thank you. :o)