(This is an example of a group in my text).
An integer 'a' has a multiplicative inverse modulo n iff 'a' and 'n' are relatively prime. So for each n > 1, we define U(n) to be the set of all positive integers less than 'n' and relatively prime to 'n'. Then U(n) is a group under multiplication modulo n.
The Attempt at a Solution
1) I am unable to wrap my mind around "An integer 'a' has a multiplicative inverse modulo n iff 'a' and 'n' are relatively prime". I know what "relatively prime" is. I just can't seem to get a grip on this, and could use a nudge in the right direction to fully grasp this group.
2) I am unsure of the identity for the group. I understand that an identity element 'e' is the element which, when multiplied by any element 'a' in the group = 'a'. So under ordinary multiplication, my identity would be 1. But this group is under multiplication modulo n, and I am unsure if the identity is something other than 1. It is the modular arithmetic component that is buggering me up.