How Do Number Theory and Group Theory Interconnect?

[dodo]

Hello,
I wonder if somebody could point me to a book (preferably), or paper, link, etc. which explores the relations between number theory and group theory.

For example, I am (more or less) following Burton's "Elementary Number Theory" and there is no mention of groups. I also have the Hardy/Wright book as a reference, and there is no mention there either.

Which is a pity, because I feel some subjects, or to put an example, Euler's totient function, or primitive roots, are better understood in the context of the multiplicative group modulo n.

 

[Bingk]

Actually, they are very strongly related, and in fact, number theory is sometimes taught with a group theory perspective ... A Classical Introduction to Modern Number Theory by K. Ireland and M. Rosen ... it's not an easy read, but it should give you a better idea of how they're related ... in essence, mod n forms the field Zn ... you can google the book, they've got a lot of the pages up, last time I checked was about 1-2 months ago :)

 

[dodo]

Hey, thanks. That book's name keeps floating around, it's not the first time I hear it; I believe it is a graduate book (I'm undergrad here). But I will have an eye on it.

 

[dodo]

Follow up: for anyone interested in this subject, here is a nice article:
http://www-math.cudenver.edu/~spayne/classnotes/subgroup.ps

As the author says, this is not original work, but a survey of existing work, for the benefit of a course. (I think his terminology is a bit flawed - what does he mean by "subgroups of Zn", when the multiplicative identity is other than 1?; but it is an interesting article anyway.)

A quick summary: we know that the coprimes to n form a multiplicative group modulo n. But, more generally, the numbers x sharing a common gcd(x,n) also form a group - only that the identity is no longer 1. But that's OK if we are not looking for subgroups of a bigger one, but for groups on their own.

For an example, if {1,3,7,9} is the multiplicative group of coprimes to 10, it also happens that {2,4,6,8}, the set of all x where gcd(x,10)=2, (a total of phi(10/2) of them), is also a group, with identity 6.

As it turns out, for any 'partition' of n into coprime factors, namely n=ab and gcd(a,b)=1, there is a group formed by the numbers x which share gcd(x,n)=a, with the identity being a^phi(b) (mod n). If you call U the group of coprimes to n, this same group is given by the set aU.

 

[CRGreathouse]

Hey, thanks. That book's name keeps floating around, it's not the first time I hear it; I believe it is a graduate book (I'm undergrad here). But I will have an eye on it.

I also recommend it. I used it as an undergrad -- though of course the class didn't finish it.

 

[Jösus]

If "A Classical Introduction to Modern Number Theory" would happen not to fit, I recommend reading "An Introduction to Number Theory", by G. Everest and T. Ward. As with the first, it is on the whole not an undergraduate-level text, but the first few chapters are not too complicated, and would probably not be too hard to get through. Additionally, from what I understood from the preface, the authors had in mind exactly what you described as being what you were looking for.
If I remember correctly they wrote that number theory often enjoyed advantages from presentations from various points of view - each of which contributes with their own insights, and so on. In the early part of the text (the part that I have read) they present very clear proofs of some different theorems in the context of elementary group and ring theory.

Perhaps this could be of interest?

 

[dodo]

Thank you, Jösus; I'm browsing it in Google Books and I'm liking what I see.