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.
 A: If you have covered elementary point set topology a possibility might be to discuss basics of topological groups. For example, show how having a (continuous) group structure on a topological space simplifies the coarsest separation axioms ($T_0$ implies Hausdorff). Not a cool theorem, but may be the first encounter with homogeneity to some of your audience.
If you want to discuss number theory and metrics, then I would consider Kronecker approximation theorem. Time permitting include the IMHO cool application: given any finite string of decimals, such as $31415926535$, there is an integer exponent $n$ such that the decimal expansion of $2^n$ begins with that string of digits
$$
2^n=31415926535.........?
$$
The downside of that is that metric properties take a back seat. You only need the absolute value on the real line and the pigeon hole principle.
A: 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.
