# Group Theory, cyclic group proof

1. Jun 1, 2010

### ripcity4545

1. The problem statement, all variables and given/known data

Prove that Z sub n is cyclic. (I can't find the subscript, but it should be the set of all integers, subscript n.)

2. Relevant equations

Let (G,*) be a group. A group G is cyclic if there exists an element x in G such that G = {(x^n); n exists in Z.}

(Z is the set of all integers)

3. The attempt at a solution

* is a binary operation, and for my purposes, is either additive (+) or multiplicative (x).

Multiplicative does not work because the multiplicative inverse of, say, 2 is not an integer. So the operation must be additive. So I can rewrite the equation for (G,+) as:

G = {nx; n exists in Z}

but that's where I get stuck. Thanks for the help!!

Last edited: Jun 1, 2010
2. Jun 1, 2010

### CompuChip

How about taking x = 1 as your generator?

3. Jun 1, 2010

### SNOOTCHIEBOOCHEE

Every cyclic group has a generator.

What is your generator in this case?

edit: nm already beaten too it

4. Jun 1, 2010

### ripcity4545

thanks to both!