Group theory with analysis I've studied group theory upto isomorphism.
Topics include : Lagrange's theorem, Normal subgroups, Quotient groups, Isomorphism theorems.
I too have done metric spaces and real analysis properly. Can you recommend any good topic to be presented in a short discussion. A good proof on an interesting problem will be highly appreciated.(E.g.- Any subgroup of (R,+) is either cyclic or dense).Is there any such problem which relates number theory and metric spaces or real analysis? 
Thanks in advance.

Complete, locally compact space, which are not discrete, and have a the algebraic structure of a commutative field with continuous operations can be classified. Look up so called "local fields" for this. They actually come with a natural metric. They are at the central interest in modern number theory. You should be able to understand the proofs given your background, but not without effort.
