"So, once you have found all the cosets whos union is the orginal group, you are done. Correct?" Well, yes...obviously!

The question here is to understand that a subgroup of a group defines an equivalence relation on the group

whose equivalence classes are precisely the left (or right) cosets of the group wrt that sbgp.

Not only that: if , so you could check easily whether

two of the sets you were given are or not different cosets.

For example, . But the cosets

are different.

Tonio