# Cyclic Subgroups of a finite group G

All elements of a finite cyclic group are of the form $a^m$, where $a^n=e$. Imagine that there were two elements $a^m$ and $a^k$ such that k is different from m, but $a^m=a^k$. Let $m=na_1+r_1$ and $k=na_2+r_2$ where the r's are the rest of the division by n and thus less than n. Lets assume, without loss of generality, that $r_1 >r_2$. Then assume $a^m=a^{r_1}$ is equal to $a^{r_2}=a^k$ and we'll get a contradiction. $a^{r_1}=a^{r_2}$ implies $a^{r_1-r_2}=e$ and thus there would be a number $r_1-r_2 such that $a^{r_1-r_2}=e$, which is a contradiction since n is the smallest natural number for which $a^n=e$.