I'm stuck on this one... I'm studying for my midterm so I'm solving problems for practice. Here's one of them... Let H be a normal subgroup in G, and let v be the natural map from G to G/H, and let X be a subset of G such that the subgroup generated by v(X) is G/H. Prove that the subgroup generated by H union X (HuX) is G. I'm trying to do this directly with showing if x is in G, then x is in <HuX> (generated subgroup). I tried doing contradiction too, by assuming <HuX> is some proper subgroup A of G and not G itself. I'm going to spend more time thinking about this. I'll be back in like 2 hours since I have a meeting, which I'll spend a minute here or there thinking about it.