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Math Help - Abstract Algebra-Groups

  1. #1
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    Exclamation Abstract Algebra-Groups

    puzzled by this problem. Suppose that a is a group element and a^(6)=e. What are the possibities for |a|?
    Intuitively I want to say that the answer is 6 because of the definition of order of an element.
    However another wants to say infinite...
    Suggestions?
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  2. #2
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    Quote Originally Posted by RoboMyster5 View Post
    puzzled by this problem. Suppose that a is a group element and a^(6)=e. What are the possibities for |a|?
    Intuitively I want to say that the answer is 6 because of the definition of order of an element.
    However another wants to say infinite...
    Suggestions?
    The order of an element, |a| , is defined as the smallest possible integer n such that a^n = e . Immediately you should notice that this rules out infinity as a possiblity as we're given that a^6=e, and clearly 6 < infinity .

    Since we're given a^6=e it must be that |a| divides 6 . This gives the possibilites:

    1) |a| = 1
    2) |a| = 2
    3) |a| = 3
    4) |a| = 6

    You can test to see if our original equation is satisfied:

    Suppose |a|= 2, then is still true that a^6 = e ? Well, a^6 = (a^2)^3 and we know that |a| = 2 and so a^2 = e

    therefore a^6=e^3=e . So it still is true. If you're not convinced you can repeat this for the other cases.

    Hope this helps.

    Pomp
    Last edited by pomp; September 30th 2009 at 04:06 PM. Reason: typo
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