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Homework Help: Subgroup of external direct product

  1. Apr 4, 2010 #1
    I am trying to do the followin 2 problems but not sure if I am doing them correct.
    Anyone please have a look....


    1. In Z40⊕Z30, find two subgroups of order 12.

    since 12 is the least common multiple of 4 and 3, and 12 is also least common multiple of 4 and 6.
    take 10 in Z40, and 10 generates a subgroup of Z40 of order 4, that is <10>={0,10,20,30}

    take 10 in Z30, 10 generates a subgroup of Z30 of order 3, that is <10>={0,10,20}

    take 5 in Z30, 5 generates a subgroup of Z30 of order 6, that is <5>={0,5,10,15,20,25}

    Answer: two subgroups of Z40+Z30 with order 12 are <(10,10)> and <(10, 5)>


    2. Find a subgroup of Z12⊕Z18 isomorphic to Z9⊕Z4.

    order of Z9⊕Z4 is 36, which is the least common multiple of 9 and 4.

    Now find a subgroup of Z12⊕Z18 with order 36.
    take 3 in Z12, 3 generates a subgroup of Z12 with order 4, that is <3>={0,3,6,9}

    take 2 in Z18, then 2 generates a subgroup of Z18 with order 9, that is <2>={0,2,4,6,8,10,12,14,16}.

    Answer: a subgroup isomorphic to Z9+Z4 is <(3,2)> in Z12⊕Z18.
     
  2. jcsd
  3. Apr 5, 2010 #2
    (1) looks good.

    In (2), your answer is correct, but all you've shown is a subgroup with the same order as Z9⊕Z4. Being isomorphic is much stronger than having the same order, though, so you're not finished on (2) yet. Try to exhibit an isomorphism, e.g.
     
  4. Apr 5, 2010 #3
    Thank you :) I will try to come up with an isomorphism from <(3,2)> to Z9⊕Z4
     
    Last edited: Apr 5, 2010
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