1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Torsion group, torsion subgroup

  1. May 9, 2009 #1
    hkhk

    if G= Z4 x Z what would be the torsion group T(G)?

    and what is the factor group of G/ T(G) ?
     
    Last edited: May 9, 2009
  2. jcsd
  3. May 9, 2009 #2

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    What have you tried?
     
  4. May 9, 2009 #3
    the cyclic group <(1,0)> is the torsion group
    ( (0,0) (1,0) (2,0) (3,0))
    ?
     
  5. May 9, 2009 #4

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    That makes no sense. From the top.. the definition of the torsion T(G) of a group G is by definition the set [itex]T(G)=\{ g\in G : g^n=e\ \mbox{for some n\in\mathbb{N}} \}[/itex] where e denotes the identity element in G. That is to say, it is simply the set of elements that have finite order!

    Do you know any group in which every element has finite order? If so, that will be a torsion group.

    Do you know any group in which no element other than the identity has finite order? If so, that will be a torsion free group.
     
  6. May 9, 2009 #5
    1. then Q is a torsion free abelian group . true?

    2. i was trying to find a set of elements of G= Z4 x Z that has finite order
     
    Last edited: May 9, 2009
  7. May 9, 2009 #6

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    What do these i) and ii) refer to?
     
  8. May 9, 2009 #7
    i edited the question
     
  9. May 9, 2009 #8

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    My post was in reference to question 1. If you can't think of a group in which every element has finite order (respectively one in which no element other than the identity has finite order), think of the groups you know and find T(G) for them.
     
  10. May 9, 2009 #9
    Z4 is a group of finite order
    and Z is not
    so Z4 is a torsion group, while Z is torsion free, am i on the right track

    sorry i do not have a very strong background in this topic and i am teaching myself this so that i can understand more advanced math
    thanks a lot for your help
     
    Last edited: May 9, 2009
  11. May 9, 2009 #10

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Well you're absolutely right: Z is torsion free since as is well known of anyone, mn=0 (m>0) cannot happen for n other than 0. And Z4 is torsion since every nonzero element in Z4 has order 4.

    As for the second question, what have you tried and where are you stuck?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Torsion group, torsion subgroup
  1. Torsion module? (Replies: 7)

  2. Groups and subgroups (Replies: 19)

  3. Curvature and Torsion (Replies: 1)

Loading...