If R is a ring (this includes the important case where R is a field), it has a simple additive group structure: just take all its elements with the addition operator. This gives an abelian group, and is boring.

We'd like to do the same with multiplication. Of course, not all of R's elements are invertible (for instance, never is!), so we define R* as the set of all invertible elements r of R (i.e. for which there exists s in R such that r s = s r = 1), equipped with the multiplication operation. This is a group, and tends to be interesting. If R is commutative, R* is abelian (this includes the case when R is a field).

Multiplicative groups modulo a number N are used extensively throughout Cryptography. I recently
had a homework assignment on multiplicative groups, and
I spent hours trying to find a decent tutorial on how
to generate them. It's actually really easy, this is what they are.

There are two types of multiplicative groups, ones
to P (a prime) or to N (just a number). For example,

Z*p, where p = 11
would be { p ... p-1 }

Z*n, where n = 12, would include all numbers up until
n-1 whose gcd(x, n) = 1, the set would be,
{ 1, 5, 7, 11 }

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