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Integral Multiples of a Set

  1. Aug 21, 2009 #1
    I'm having a bit of a tough time interpreting <S> for a set S. I know for an element a, <a> is the set of all integral powers of a with respect to a given operation, but for a set S = {a, b, c}, what would <a, b, c> turn out as?

    Edit: The source of my trouble is with this: The subgroup <9, 12> of the group of integers with addition as the operation contains 12 + (-9) = 3 (in order for it to be a group). Here is what the text says: "Therefore <9, 12> must contain all multiples of 3." I thought <9, 12> would only consist of multiples of 9 and 12, but apparently, there is more to it.
     
    Last edited: Aug 21, 2009
  2. jcsd
  3. Aug 21, 2009 #2
    Not knowing more about the group in question, I am assuming that the group operation is addition. Then [itex]\langle 9, 12 \rangle[/itex] is going to be all linear combinations of 9 and 12, i,.e. [itex]9m+12n[/itex] for integers m and n. It turns out that that this is identical to the set of all multiples of 3.

    I suspect that for any two intgers a and b that [itex]\langle a,b \rangle = \langle \gcd (a,b) \rangle[/itex].

    --Elucidus
     
  4. Aug 22, 2009 #3
    Thanks for the reply...it seems as though it never came to me anywhere in the text. I was thinking the notation for <a1, a2 ... an> was simply the set of all integral powers of the elements.

    So from this, I'm assuming that < > with respect to addition can be interpreted as a linear combination between elements in the group, right?
     
  5. Aug 22, 2009 #4
    Are you sure < > notation was never defined in that book? From your comments, I am guessing that (for that book) <S> means the subgroup generated by S.
     
  6. Aug 22, 2009 #5
    Yup...that was part of its definition. The definition it gave was: <a> is the set of all integral powers of a for a given operation. They then went into further analysis. I just wasn't sure what <S> of a set S = {a, b, c, ...} was since the definition they gave was for a single element a.
     
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