Results 1 to 7 of 7

Math Help - tensor product

  1. #1
    Newbie
    Joined
    Aug 2010
    Posts
    24

    tensor product

    If G is finite Abelian group and G (X)_Z Z/pZ={0} ( tensor of G with Z/pZ over Z )
    for all primes p, then show that G = {0}. Does the result remain true if G is infinite ?

    Since G is finite abelian then G= Z/n_1Z x Z/n_2Z x ... x Z/n_kZ
    where n_k | n_k-1 | ... | n_2 | n_1
    so {0}= G(X)Z/pZ = Z/dZ where d=gcd(n_k , p) for all primes
    then d=1 and for each p there is one of n_i relatively prime to p
    but can this imply G is zero ?
    Also, I have no idea about the second assertion.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by hgd7833 View Post
    If G is finite Abelian group and G (X)_Z Z/pZ={0} ( tensor of G with Z/pZ over Z )
    for all primes p, then show that G = {0}. Does the result remain true if G is infinite ?

    Since G is finite abelian then G= Z/n_1Z x Z/n_2Z x ... x Z/n_kZ
    where n_k | n_k-1 | ... | n_2 | n_1
    so {0}= G(X)Z/pZ = Z/dZ where d=gcd(n_k , p) for all primes
    then d=1 and for each p there is one of n_i relatively prime to p
    but can this imply G is zero ?
    Also, I have no idea about the second assertion.

    The second assertion is easy to show is false: take \mathbb{Q}\otimes_\mathbb{Z} G , with G any finite abelian group, say of order

    n\Longrightarrow q\otimes g=n(q/n\otimes g)=q/n\otimes ng=q/n\otimes 0=0 ...

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2010
    Posts
    24
    No. The question is : what if G is infinite , you're assuming G is finite in your argument.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by hgd7833 View Post
    No. The question is : what if G is infinite , you're assuming G is finite in your argument.

    No, \mathbb{Q} is infinite and Z-tensored with any finite abelian group, and in particular with \mathbb{Z}/p\mathbb{Z} we get the zero group, thus showing that an infinite group can give zero tensored with any group of order a prime p and nevertheless it is not, obviously, the zero group.

    Tonio
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Aug 2010
    Posts
    24
    Oh yes, so Q is your group. This works now.
    Did you look at my argument for the first part ? Is it correct ? Thanks
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    for the first part, suppose to the contrary that G \neq \{0\}. then |G| \geq 2 and \{0\}=G \otimes \mathbb{Z}/p \mathbb{Z} \cong \bigoplus_{i=1}^k (\mathbb{Z}/n_i \mathbb{Z} \otimes \mathbb{Z}/p \mathbb{Z}). thus \mathbb{Z}/n_i \mathbb{Z} \otimes \mathbb{Z}/p \mathbb{Z}=\{0\}, for all i and all primes p.

    in particular \mathbb{Z}/n_1 \mathbb{Z} \otimes \mathbb{Z}/p \mathbb{Z}=\{0\} for all primes p. but that is false because if p \mid n_1, then \mathbb{Z}/n_1 \mathbb{Z} \otimes \mathbb{Z}/p \mathbb{Z} \cong \mathbb{Z}/p\mathbb{Z} \neq \{0\}.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Aug 2010
    Posts
    24
    Yes I got it now. Thanks very much to you and to Tonio !
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Tensor Product of Modules
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 9th 2011, 08:49 AM
  2. Tensor product and basis
    Posted in the Differential Geometry Forum
    Replies: 13
    Last Post: January 24th 2011, 03:47 PM
  3. example, tensor product
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 12th 2010, 12:41 PM
  4. Tensor product constructs
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 17th 2010, 04:22 PM
  5. Verification of tensor product
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 3rd 2009, 07:17 PM

Search Tags


/mathhelpforum @mathhelpforum