math - 乘法阶数与乘法组的阶数

Question

如何证明所有乘法阶数除以F13的乘法群F的阶数（大小）。.

Answer 1

由于该群是阿贝尔群，因此最简单的方法是使用任何元素的乘法是双射。令 F = {g1, g2, g3, ..., gn} 并令 h 为任意元素。然后还有 F = {h*g1, h*g2, ..., h*gn}。因此，将所有元素相乘，我们得到 g1 * g2 * g3 * ... * gn = h*g1 * h*g2 * ... * h*gn。但后者等于 h^n * g1 * g2 * ... * gn。现在使用取消定律得出结论 h^n = 1，由此得出结果。

Answer 2

You show that the cyclic group <x> generated by an element x is a subgroup of IF* and that "u~v iff u^(-1)*v in <x>" is an equivalence relation that divides the multiplicative group into equivalence classes of equal size.

So that you get

[size of IF*] 
     = [size of <x>] * [number of equivalence classes]

which means that the order of x = [size of <x>] is a divisor of the number of invertible elements, i.e., the size of the multiplicative group of IF

See also the little theorem of Fermat.