[robryk]

I think you are on the right track (sorry, I did not verify all the individual statements). If you weren't already aware of them, you might wish to learn what normal subgroups are, see how you can have a subgroup that's not a normal subgroup (and probably see what cosets are at the same time), and see how does dividing a group by a normal subgroup (to yield a group) work and what properties it has.