Multiplicative group modulo n

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Definition

Let n be a positive integer. The multiplicative group modulo n is the subgroup of the multiplicative monoid modulo n comprising the elements that have inverses.

Equivalently, it is the group, under multiplication, of elements in \{ 0,1,2,\dots,n-1\} that are relatively prime to n. (The two definitions are equivalent because if a and n are relatively prime, there exist integers x,y such that ax + ny = 1, so ax \equiv 1 \mod n).

Facts

  1. The order of the multiplicative group modulo n equals the number of elements in \{ 0,1,2,\dots, n-1\} that are relatively prime to n. This number is termed the Euler-phi function or Euler totient function of n, and is denoted \varphi(n).
  2. For a prime p, \varphi(p) = p - 1. In other words, every nonzero element less than p is invertible modulo p.
  3. The multiplicative group modulo n is a cyclic group if and only if n = 2,4,p^k,2p^k for p an odd prime and k a natural number.