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Find the subgroup!

  1. Dec 6, 2004 #1
    let p and q be distinct primes. suppose that H is a proper subset of integers and H is a group under addition that contains exactly 3 elements of the set
    { p,p+q,pq, p^q , q^p}.
    Determine which of the foll are the 3 elements in H
    a. pq, p^q, q^p

    b. P+q, pq,p^q

    c. p, p+q, pq

    d. p, p^q, q^p

    e. p, pq, p^q
  2. jcsd
  3. Dec 6, 2004 #2

    matt grime

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    Hint Euclid's algorithm: p^r and q^s are coprime so if H contains these two elements, then it contains 1, and hence is Z. Use this idea in several variations. Of course you could consider the group pZ
  4. Dec 7, 2004 #3
    thanks sir, but could you please elaborate further.
    i don't seem to get the idea...
  5. Dec 7, 2004 #4

    matt grime

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    If a group contains p, it contains np for all n in Z. So clearly e. forms the answer.

    A group for instance cannot contain p and q if they are coprime and not be all of Z since there are a and b in Z such that ap+bq=1, hence the group contains all elements of Z.

    And I tihnk you ought to ponder that for a while, cos I really have given you more information than I want to.
  6. Dec 7, 2004 #5

    matt grime

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    How about thinking about an example if you cant' see it:

    p=2 q=3

    If 2 and 3 are in the group, then so is -2 (inverses) and hence, so is 3-2=1 (composition)

    If 1 is in there so is 1+1+1+..+1= n (composition) and n was arbitrary, also -n is in there (inverses again)
  7. Dec 12, 2004 #6
    well...thanks a lot, sir!! i figured it out...
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