For a coprime group of N, it looks to me that every element can multiply a positive integer up to N+1 then mod N to generate the whole original coprime group. Is this true? If so, what property is this called?
For example, N = 15, and element of coprime group N will be {1,2,4,7,8,11,13,14}. I have tried all elements and it works. For example, for element 4:
4 * 2 mod 15 = 8
4 * 4 mod 15 = 1
4 * 7 mod 15 = 13
4 * 8 mod 15 = 2
4 * 11 mod 15 = 14
4 * 13 mod 15 = 7
4 * 14 mod 15 = 11
4 * 16 mod 15 = 4
So it generates the whole original coprime group N. It seems this should be some property, but I couldn't find the answer.
Euler employs this fact as the key lemma in his proof of the totient theorem.
Left multiplication by elements of a group always permutes the elements of the group,. It defines a group action. So you actually get a homomorphism from $G$ to $\operatorname{\\Sym}G$.
This is how Cayley's theorem, which says that every group can be embedded in a symmetric group, is done.
Since it works for any $G$, it works for the group of units, $\Bbb Z_n^×$.