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Number theory and groups

  1. Oct 25, 2009 #1
    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.
  2. jcsd
  3. Oct 25, 2009 #2
    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 alot of the pages up, last time I checked was about 1-2 months ago :)
  4. Oct 25, 2009 #3
    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.
  5. Nov 10, 2009 #4
    Follow up: for anyone interested in this subject, here is a nice article:
    http://www-math.cudenver.edu/~spayne/classnotes/subgroup.ps [Broken]

    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.
    Last edited by a moderator: May 4, 2017
  6. Nov 10, 2009 #5


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    I also recommend it. I used it as an undergrad -- though of course the class didn't finish it.
  7. Dec 2, 2009 #6
    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?
  8. Dec 3, 2009 #7
    Thank you, Jösus; I'm browsing it in Google Books and I'm liking what I see.
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