Results 1 to 6 of 6
Like Tree1Thanks
  • 1 Post By HallsofIvy

Thread: The integers modulo m with zero removed are a cyclic group for multiplication modulo

  1. #1
    Newbie
    Joined
    Dec 2018
    From
    BUENOS AIRES
    Posts
    8

    The integers modulo m with zero removed are a cyclic group for multiplication modulo

    Hi: This is An Introduction to the Theory of Groups by George Polites, example 11, page 9: please see first attachment below. And now he gives this exercise: please see second attachment below.

    I say any prime will do. If m is a prime then G will be a field for
    sum and product modulo m. In a field the elements different from zero are
    a group for multiplication and this group is cyclic. I remember it was not a trivial matter to
    prove that group is cyclic. How can the author put such a difficult
    problem at such an elementary stage? He has just shown Lagrange theorem!

    The fact is I cannot solve the exercise. I chose this book because I hoped
    it would be easy but maybe I am wrong.
    Attached Thumbnails Attached Thumbnails The integers modulo m with zero removed are a cyclic group for multiplication modulo-att1.png   The integers modulo m with zero removed are a cyclic group for multiplication modulo-att2.png  
    Last edited by STF92; Dec 10th 2018 at 03:22 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    20,214
    Thanks
    3340

    Re: The integers modulo m with zero removed are a cyclic group for multiplication mod

    If m is not prime then we can factor m= pq where p and q are also integers. Both p and q are less than m of course so in $\displaystyle Z^m$. What is pq (mod m)? What does that tell you about the multiplicative inverses of p and q in this group? Do you see that this cannot happen if m is prime?
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Dec 2018
    From
    BUENOS AIRES
    Posts
    8

    Re: The integers modulo m with zero removed are a cyclic group for multiplication mod

    Is $Z^m$ the multiplicative group ${1, 2, 3,...,m-1}$?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Dec 2018
    From
    BUENOS AIRES
    Posts
    8

    Re: The integers modulo m with zero removed are a cyclic group for multiplication mod

    pq= bm + r, $0<= r < m$. pq is r. Or better said, pq = 0 mod m (not pq = 1) in which case $p^{-1} = q$. If m is prime I cannot factor. I do not quite understand.

    EDIT: the exersize statement is misleading. I should find conditions on m to make G a group. The author says a CYCLIC group. If it is a group of course it will be cyclic. Every i in G should be prime relative to m. In this way i will have an inverse in G. To achieve this, m must be prime. That's all.
    Last edited by STF92; Dec 10th 2018 at 09:06 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Apr 2005
    Posts
    20,214
    Thanks
    3340

    Re: The integers modulo m with zero removed are a cyclic group for multiplication mod

    Quote Originally Posted by STF92 View Post
    Is $Z^m$ the multiplicative group ${1, 2, 3,...,m-1}$?
    That is only a group if m is prime.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Dec 2018
    From
    BUENOS AIRES
    Posts
    8

    Re: The integers modulo m with zero removed are a cyclic group for multiplication mod

    I agree ("I say any prime will do.", post #1). How do I mark the thread solved?
    Last edited by STF92; Dec 12th 2018 at 02:19 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Multiplication Modulo n on Z_m
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Oct 15th 2012, 06:39 AM
  2. [SOLVED] Modulo Arithmetic, set of all integers div 7
    Posted in the Discrete Math Forum
    Replies: 8
    Last Post: May 8th 2011, 09:57 AM
  3. Cyclic groups modulo
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Mar 24th 2011, 09:39 AM
  4. Modulo Definition of Addition and Multiplication
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Mar 16th 2011, 08:22 AM
  5. Integers modulo n
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Jul 8th 2008, 02:03 PM

/mathhelpforum @mathhelpforum