How can I show all elements of a cyclic subgroup are distinct?
All elements of a finite cyclic group are of the form , where . Imagine that there were two elements and such that k is different from m, but . Let and where the r's are the rest of the division by n and thus less than n. Lets assume, without loss of generality, that . Then assume is equal to and we'll get a contradiction. implies and thus there would be a number such that , which is a contradiction since n is the smallest natural number for which .
Now do the infinite cyclic one.