Modular Mathematics, Permuting the Units

Permuting the Units

The concepts of prime and composite don't make much sense in modular mathematics, but we still retain the notion of a unit. Let x be a unit mod m if there is some y such that xy = 1 mod m. In other words, y is the inverse of x. Stepping back to the integers, we are asking whether yx - km = 1 has any solutions. In fact it has an entire lattice of solutions in y and k, iff x and m are relatively prime. Each value of y advances by m, hence all values of y are the same mod m, and the inverse of x is unique.

If u has inverse v and x has inverse y, ux has inverse vy. Thus the product of units is another unit. Multiplication by u maps units to units, and the map can be reversed by multiplying by v. The map is one to one and onto, permuting the units about.