Q: Any use of advanced Abstract Algebra in Differential Geometry? I believe that if someone is going to continue their studies and doing research on Differential Geometry's topics, would never need advanced Abstract Algebra (or maybe not even undergraduate level of it). But why in any graduate program, students must take advanced Abstract Algebra (for example the book Algebra by Hungerford) as a compulsory and common course for any pure mathematics program?
I don't think even advanced Abstract Algebra is needed for Real Analysis (?)       

A: You will most likely be involved in teaching as well as research in all but a few cases.  And many fields can overlap considerably!  These core areas vary from school to school, but as someone with a graduate degree in math, you are generally expected to know something from various subjects and not just your own.  For example, I am a discrete mathematician, and haven't really used analysis much even though I was required to take it.  But I have taught Calculus several times, and I'm expected to have a rigorous understanding of the subject even if I'm not teaching delta-epsilon proofs.
Also as the comments have suggested, you will come across quite a bit of algebra in studying certain aspects of differential geometry.  There are quite a few connections to algebraic geometry, even an area called algebraic differential geometry.
