How to study abstract algebra I am taking Abstract Algebra course at the university. We are doing chapters 1-20 from Gallian's abstract algebra text book.
I am just doing assigned homework everyweek ( About 5 questions from each chapter). Although I am getting an A in all the assignments and midterms, but I am really worried that my understanding might be shallow or just enough to do the homework that I am going to promptly forget when the course is over. 
My question is how do I know that I am gaining knowledge that would stay with me and not just studying enough to do the assignments.
Also, What else can I do other than just doing homework assignments. 

Some random points that may help in any math class:


*

*Solve more problems not only the ones assigned. Take problems from other textbooks.

*Try to explain the material to some of your colleagues that maybe having difficulties with the course. Tutoring someone is a great way to discover points you thought you understood but you really do not have a firm grasp yet.

*For every new definition you see, come up with a list of concrete examples.

*For every theorem you see, remove one of its assumptions and try to get a counter-example.

