Results 1 to 2 of 2

Math Help - Quotient groups

  1. #1
    Junior Member
    Joined
    Mar 2009
    Posts
    42

    Quotient groups

    Let a, b, c, d be integers, and let H be the subgroup of Z Z generated by (a, b) and (c, d). Thus H is the set of all elements of the form (ma, mb) + (nc, nd) where m, n ∈ Z.


    (a) Suppose b = 0, c = 0. Prove that (Z Z)/H is isomorphic to Z/aZ Z/dZ.

    (b) Suppose (a, b) = (10, 12) and (c, d) = (4, 4). Find a product of cyclic groups isomorphic to (Z Z)/H .

    (c) Determine, in terms of a, b, c, d, when (Z Z)/H is finite. When it is finite, find its order.


    I can see intuitively that the statements for a and b are right but I can't come up with any method of proof
    could anyone help me?? plelase ?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie lepton's Avatar
    Joined
    Sep 2009
    Posts
    10
    For (a), construct a surjective (onto) homomorphism from \mathbb{Z}\times \mathbb{Z} to \mathbb{Z}/a\mathbb{Z}\times \mathbb{Z}/b\mathbb{Z} whose kernel is H, then apply the First Isomorphism Theorem for groups.

    For (b), recall that 'modding' out by H just means we impose the relation (10m+4n,12m+4n)=(0,0) in \mathbb{Z}\times\mathbb{Z}. In which subgroup of \mathbb{Z} does 10m+4n=0 for all m,n? 12m+4n=0? The product of these two groups is what you are looking for.

    For (c), look at gcd(a,b) and gcd(c,d), and recall that the gcd(x,y) can always be written as a linear combination of x and y.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Quotient Groups - Infinite Groups, finite orders
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: August 11th 2010, 07:07 AM
  2. Quotient Groups
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: May 4th 2010, 03:30 PM
  3. Help with Quotient Groups
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: November 24th 2009, 12:08 AM
  4. Quotient Groups
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: November 17th 2009, 04:42 AM
  5. quotient groups
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: September 22nd 2009, 03:19 AM

Search Tags


/mathhelpforum @mathhelpforum