Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Groups of homomorphisms of abelian groups

  1. Oct 10, 2014 #1
    Hello everybody!
    I've just started with studying group homorphisms and tensor products, so i am still not very sure if i undertstand the subject correct. I am stuck with a question and i would ask you for some help or hints how to proceed...

    What i have to do is to describe ##Hom(\mathbb{Q}/\mathbb{Z},\mathbb{Q})## and ##Hom(\mathbb{Q},\mathbb{Q}/\mathbb{Z})##. I know that both ##\mathbb{Q}/\mathbb{Z}## and ##\mathbb{Q}## are abelian groups, ##\mathbb{Q}/\mathbb{Z}## as a ##\mathbb{Z}##-module is finitely generated, but ##\mathbb{Q}## as a ##\mathbb{Z}##-module is not finitely generated.
    Can anybody help me with this problem? How is it meant "to describe the groups of homomorphisms"?
    Thank you in advance!
     
  2. jcsd
  3. Oct 12, 2014 #2

    lavinia

    User Avatar
    Science Advisor
    Gold Member

    A couple of things that may help
    (Since Z is not a group under multiplication, you must be referring to addition)

    - Q/Z is a torsion group. That is: every element is of finite order. Q has no elements of finite order.

    - Try to find a set of Z generators of Q. Do the reciprocals of the prime numbers work?


    BTW: Why do you think Q/Z is a finitely generated Z module?
     
    Last edited: Oct 12, 2014
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Groups of homomorphisms of abelian groups
  1. Group homomorphism (Replies: 7)

  2. Group homomorphism (Replies: 4)

Loading...