Suppose that $a$ has order $15$. Find all of the left cosets of $\langle a^5\rangle $ in $\langle a\rangle$ .
Ok, so I know by Lagrange's Theorem, that the order of the subgroup divides the order of the group. Therefore, the index of the cosets must be $3$.
However... How do I apply this index to the subgroups? I know the final answer, I just want the breakdown, in a meaningful way. The text I use did not really elaborate on this with the notation given. I think I may just be confused for that reason.