What is a generator in Zn star?

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Basically, an element, g, of a cyclic group is a generator if you can get every element of the group by just taking powers of g.

I'm not sure of an "easy" way to find them, other than brute force. Just take an element and compute its order. If you generate the entire grouo along the way, you have a generator.

For example, Z/7Z *

Take the element 3: (remember mod 7)

3^1 = 3

3^2 = 2

3^3 = 6

3^4 = 4

3^5 = 5

3^6 = 1

Since we hit the identity, we are done. Notice 3 generated every element in Z/7Z* which tells us it is a generator.

There are theorems that tell us how many generators to expect to have under special circumstances, but none that I know of that just give the generators to you.

Of which? Which course is this btw?

A generating set is a set of elements whose products with each other generate the group.

Usually, we look for the smallest set that satisfies this.

In the case of cyclic groups, the smallest generating set will just be the generator, so it will have one element.

Basically, you can imagine trying to extend the idea of a generator to any, non-cyclic group and ask,

"Okay, there isn't just one element that generates the group, but this one was really close. How about I throw in another element, and do some products that gets me the elements I was missing from the original group?."

For example, the group of integers under addition has generating set S={-1,1} since every element can be generated by multiple addition of -1 or 1.