abstract algebra - groups?

Question

Consider the group Q (rational numbers) under addition. Note that Z (integers) is normal in Q, and therefore Z\Q is a group. Show that this group is infinite but that each element of the group has finite order.

I can see that this is true, but I don't know how to prove it.

Answer 1

Take a typical element in Z\Q, say Z+(m/n) where m/n is in lowest terms. Now n*(m/n)=m which is in Z, which means the coset (or element in Z\Q) has finite order.

To show that Z\Q is infinite, you just need that there are infinitely many elements in Z/Q, which means there are infinitely many fractions of the type 1/n, where n not equal to zero a positive interger.