I've learned that the commutator subgroup is generated by the commutators. Now this says little about its elements (to me) because I don't see how they need to be commutators themselves. I'm interested in what these commutators look like in general (maybe an intuitive idea). And what some easy ways of proving that something equals a commutator subgroup.
I'm also quite confused about the difference of normal subgroups and conjugacy classes. I've read that normal subgroups are usually the 'union' of conjugacy classes, however.. Can't we construct normal subgroups then by taking individual conjugacy classes?