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

Homework Help: Coset Question

  1. Feb 12, 2012 #1
    Let H and K be Subgroups
    show Ha[itex]\cap[/itex]Ka = (H[itex]\cap[/itex]K)a for all a [itex]\in[/itex]G

    pf

    Let x[itex]\in[/itex]Ha[itex]\cap[/itex]Ka

    Then x[itex]\in[/itex]Ha and x[itex]\in[/itex]Ka

    Can I just say that x [itex]\in[/itex](H[itex]\cap[/itex]K)a ? Or am I missing something.
     
  2. jcsd
  3. Feb 12, 2012 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    hi Punkyc7! :smile:
    nooo :redface:

    your next word should be "∃" :wink:
     
  4. Feb 12, 2012 #3
    huh?
     
  5. Feb 12, 2012 #4

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    sorry, that character doesn't show up on some computers :redface:

    your next words should be "there exists a y such that …" :smile:
     
  6. Feb 12, 2012 #5
    Ok so there exist a y such that y is in Ha and Ka. Then is it right?
     
  7. Feb 12, 2012 #6

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    no!!

    x is in Ha and Ka, so there exists a y in … such that … ? :smile:
     
  8. Feb 12, 2012 #7
    bear with me..

    There exist a y in (H[itex]\cap[/itex]K)a such that x=ya?
     
  9. Feb 12, 2012 #8

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    not quite

    read it and try again :smile:
     
  10. Feb 12, 2012 #9
    is it a y in G? such that x=ya? Then im not sure if I need the a anymore.
     
  11. Feb 12, 2012 #10

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    no!

    once again :smile:
     
  12. Feb 12, 2012 #11
    ok Im sure where the y is floating around but if its not in G or (H[itex]\cap[/itex]k)a. The only place left would have to be just H[itex]\cap[/itex]K right
     
  13. Feb 12, 2012 #12

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    yes!!!!! :smile:

    now, can you see why?
     
  14. Feb 12, 2012 #13

    Deveno

    User Avatar
    Science Advisor

    suppose y is in both Ha and Ka.

    what does this mean?

    it means y = ha, for some h in H, and y = ka, for some k in K.

    for this h, and this k:

    ha = ka.

    can you think of something to do with this? (perhaps multiply both sides by something?)

    *****

    that is only HALF the problem, though. the other half means you suppose:

    y is in (H∩K)a.

    what can you do with this?
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook