Properties Of An Element With Infinite Order

Suppose the order of $a$ is infinite. Then by definition this means that $a^n\neq e$ unless $n=0$. Clearly if $i=j$ then $a^i=a^j$. We only need to prove the converse of this statement. Suppose that $a^i=a^j$, and assume without loss of generality that $i\geq j$. Then $a^{i-j}=e$ (just multiply both sides on the right by $a^{-j}$). Because $a$ has infinite order, then we know $a^k\neq e$ for any positive integer $k$. Since $i-j\geq 0$ and $a^{i-j}=e$, we must have $i-j=0$. This means $i=j$.