How can I show all elements of a cyclic subgroup are distinct?

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- September 10th 2011, 09:39 AMdwsmithCyclic Subgroups of a finite group G
How can I show all elements of a cyclic subgroup are distinct?

- September 10th 2011, 11:05 AMModusPonensRe: Cyclic Subgroups of a finite group G
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.