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Math Help - Algebra: abelian finite group

  1. #1
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    Question Algebra: abelian finite group

    12 i) Let A be an abelian finite group, and let a,b be elements in A.
    Show that the order of ab is a divisor of the lcm( (ord(a), ord(b) ).

    Okay, I tried:
    It follows by definition that all elements in A have finite order.
    lcm( (ord(a), ord(b) ) = x, so x divides ord(a) and x divides ord(b).

    I have to show that the order of ab is a divisor of x.
    My first thought:
    I have to show that "the order of ab" is a divisor of "x".
    Since "x" divides both ord(a) and ord(b) it follows:
    ord(a) / x / order of ab = ord(a) / (x*order of ab)
    ord(b) / x / order of ab = ord(b) / (x*order of ab)
    Well, I guess I'm going the wrong way because it doesn't make sense.
    Can someone help me get further?
    Tnx. & sorry if I got the topic ("pre-algebra and algebra" wrong!)


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  2. #2
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    Let's try some examples:

    if a^5=1 and b^7=1, then clearly (ab)^{35}=1.

    if a^4=1 and b^6=1, then clearly (ab)^{12}=1.

    You're on the right track, it's just that ord(a) | lcm(ord(a),ord(b)), not the other way around.

    Also, I think it should be ord(ab) = lcm(ord(a),ord(b)). Do you have an example where ord(ab) < lcm(ord(a),ord(b))?
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  3. #3
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    Quote Originally Posted by qmech View Post
    You're on the right track, it's just that ord(a) | lcm(ord(a),ord(b)), not the other way around.

    Also, I think it should be ord(ab) = lcm(ord(a),ord(b)). Do you have an example where ord(ab) < lcm(ord(a),ord(b))?
    Hm, I don't really get what you're trying to show me here. Like; why do you think ord(ab) = lcm(ord(a),ord(b)) ? What's that got to do with it?

    Ill try.
    ord(ab) = lcm(ord(a),ord(b))
    I have to show that ord(ab) | lcm((ord(a), ord(b))
    And yeah, if ord(ab) is lcm((ord(a), ord(b)), it's easypeasy, because x divides x, so that's the proof that ord(ab) | lcm( (ord(a), ord(b)).
    But then, how to show that ord(ab) = lcm(ord(a), ord(b)) ?
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  4. #4
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    I think I can use Lagrange's theorem, but how? I have no idea.
    Can anyone help me please?
    Last edited by MaryB; December 6th 2009 at 04:55 AM.
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