Recall that a möbius function has 4 parameters; they are nicely arranged in 2 rows (numerator and denominator) and in 2 columns (x coefficient and unit coefficient). So it looks nice to arrange them in a 2x2 matrix. Wouldn't it be even nicer if the matrix made sense?

It does!

The trivial case for these functions is when the denominator divides the numerator, and the function is constant. This occurs precisely when the matrix is singular. In all other cases, the function is one to one.

Now, consider what happens when you compose the möbius functions fA and fB represented by 2 matrices A and B. The function f(x) = fA(fB(x)) is also a möbius function (verify this by substitution). And even better -- the coefficients of f are exactly given by the matrix A B!

Note, however, that the correspondence to 2x2 matrices is not one to one: if we multiply both numerator and denominator by the same constant, we change the matrix but not the function. So the group of möbius functions is a quotient group of GL2, but not isomorphic to it. However, if we are in C we can always multiply numerator and denominator by a constant to get determinant 1, so we're looking at PSL2(C). If we're in R, we can always get the determinant to be +1 or -1, so we're looking at SL2(R). And if we're in Z, we're basically looking at some number theory to decide anything, but it turns out that all the interesting functions are in SL2(R) anyway.

For real coefficients, the sign of the determinant is also important.

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