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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.

UNM
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    What does "study group theory but isomorphism" mean? You've studied group theory...but not what isomorphism is? But then you mention isomorphism theorems...?? – DonAntonio Sep 13 '13 at 13:45
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    You could look up "Proofs from THE BOOK". There are a number (well, I can only remember one, so there is may be only one...) of proofs which use group theory to prove interesting results. For example, there is a proof which uses group theory to prove that there are infinitely many primes. Which is actually kinda cool! – user1729 Sep 13 '13 at 14:19
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    It's not exactly clear what you want. What kind of discussion is that topic for? – Patrick Da Silva Sep 13 '13 at 15:31
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    You could talk about something like the $p$-adic numbers (a metric space, related to number theory) and explain Hensel's lemma--this is a version of "Newton's method" from calculus in that setting. – tkr Sep 13 '13 at 15:31
  • Any cool kind of proof like our friend said "we can prove that there are infinitely many primes" – UNM Sep 13 '13 at 15:33
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    Ask your advisor. – Potato Sep 13 '13 at 15:33
  • @Potato It is not clear that there is an advisor. This is not a "give me a topic for my dissertation" question. – user1729 Sep 13 '13 at 16:24
  • As @PatrickDaSilva said It is not clear what kind of student paper you are about to write. How much time are you willing to spend on the paper? – AD - Stop Putin - Sep 14 '13 at 07:51
  • Did you read set theory? Did you know that every non-empty set can be made into a group? – AD - Stop Putin - Sep 14 '13 at 16:17

2 Answers2

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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.

Community
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Jyrki Lahtonen
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  • I picked out these topics partly because while they are well known and not deep, they are not necessarily encountered in any normal undergraduate course. That kind of topics make for fine term papers and such as they are quick to get into, don't place any very specific demands on the student's background, and aren't too likely be covered in a course the student may take later. – Jyrki Lahtonen Sep 14 '13 at 07:36
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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.

Marc Palm
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