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Thread: verify the multiplication of two absolute abelian group elements.

  1. #1
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    verify the multiplication of two absolute abelian group elements.

    **Question**
    Given that a and b belongs to an Abelian Group and |a|=6 and |b|=6, whay can you say about |ab|?


    **My solution**
    |a|=6
    a=6
    a^6=e


    similarly;
    |b|=6
    b=6
    b^6=e


    hence
    |ab|=(ab)
    (ab)=(ab)^6
    =a^6b^6
    =ee
    =e


    therefore |ab|=6 and ab belongs to G


    Q; Pls what sis I do wrong in answering the question???


    Thanks
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  2. #2
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    Re: verify the multiplication of two absolute abelian group elements.

    First, is |a| the order of element a- that is, the power, n, to which a must be raised so that a^n= e? If so what you are doing is correct except where you say "|ab|= (ab)" and (ab)= (ab)^6". I don't know what those could mean. |ab| is, of course, the order of ab but that is a number but what is (ab)? If you just mean "a times b" it is not equal to a number as |ab| is. And I cannot see any interpretation which makes "(ab)= (ab)^6" true. It is true, since G is abelian, that (ab)^6= a^6b^6= 6. To complete the proof that |ab|= 6 you would also need to show that no number less than 6 works. Of course, you do not need to say "and ab belongs to G"- that was given when saying that a and b are both in G.
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  3. #3
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    Re: verify the multiplication of two absolute abelian group elements.

    As you point out, $(ab)^6=e$. So the order of ab is a divisor of 6, namely 1, 2, 3 or 6. Another question arises; is each of these orders actually possible? That is, can you find a group where each possibility is realized? The answer is yes. Hint: look in the direct product of 2 cyclic groups of order 6.
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  4. #4
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    Re: verify the multiplication of two absolute abelian group elements.

    |ab|=?
    (ab)^6 =a^6b^6
    =ee
    =e
    hence if
    a^n = e
    a^6 = e

    same applies to b;


    where |e| = 1.

    hence, (ab)^n = e


    as above, (ab)^6 = e..


    so what does |ab| stand for then

    is |ab| = 1??????
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