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Math Help - Abelian group

  1. #1
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    Abelian group

    a) Prove that if a and b are elements of an Abelian group G, with o(a) = m and o(b) = n, then (ab)^{mn}= e .

    b) With G, a, and b as in part (a), prove that o(ab) divides o(a)o(b).

    c) Give an example of an Abelian group G and elements a and b in G such that o(ab) \neq\o(a)o(b). Compare part (b)

    I suspect I could figure out, or at least know where to start, part b if I knew part a (If t is an integer, then a^t=e iff n is a divisor of t)

    But, I just don't know what to do with part a...

    From what I understand, a^m=e and b^n=e

    I have done all sorts of random inserts of a^m, a^m * a^-m , etc. Nothing seems to give me what I need though... erm, I was wondering, since a^m=e does a^{-m}=e?
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  2. #2
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    Quote Originally Posted by Zorae View Post
    a) Prove that if a and b are elements of an Abelian group G, with o(a) = m and o(b) = n, then (ab)^{mn}= e .

    b) With G, a, and b as in part (a), prove that o(ab) divides o(a)o(b).

    c) Give an example of an Abelian group G and elements a and b in G such that o(ab) \neq\o(a)o(b). Compare part (b)

    I suspect I could figure out, or at least know where to start, part b if I knew part a (If t is an integer, then a^t=e iff n is a divisor of t)

    But, I just don't know what to do with part a...

    From what I understand, a^m=e and b^n=e

    I have done all sorts of random inserts of a^m, a^m * a^-m , etc. Nothing seems to give me what I need though... erm, I was wondering, since a^m=e does a^{-m}=e?

    The trick is that in an abelian group, (ab)^m=a^mb^m...as simple as that!
    Tonio
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  3. #3
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    Quote Originally Posted by Zorae View Post
    a) Prove that if a and b are elements of an Abelian group G, with o(a) = m and o(b) = n, then (ab)^{mn}= e .

    b) With G, a, and b as in part (a), prove that o(ab) divides o(a)o(b).

    c) Give an example of an Abelian group G and elements a and b in G such that o(ab) \neq\o(a)o(b). Compare part (b)

    I suspect I could figure out, or at least know where to start, part b if I knew part a (If t is an integer, then a^t=e iff n is a divisor of t)

    But, I just don't know what to do with part a...

    From what I understand, a^m=e and b^n=e

    I have done all sorts of random inserts of a^m, a^m * a^-m , etc. Nothing seems to give me what I need though... erm, I was wondering, since a^m=e does a^{-m}=e?
    1) a^m=e, b^n=e , which follows a^{mn}=e, b^{mn}=e. Since G is abelian, (ab)^{mn}=a^{mn}b^{mn}=e .
    2) Let o(a)=m, o(b)=n. By using the basic number theory property, we see that mn=gcd(m,n)lcm(m,n). We know that a^{lcm(m,n)}=e and b^{lcm(m,n)}=e. Since G is abelian, it follows that (ab)^{lcm(m,n)}=e. Thus o(ab) | lcm(m,n) and o(ab)|mn.

    3) Check 2 and 3 in \mathbb{Z}_4.
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