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Thread: Problem with order of element

  1. #1
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    Problem with order of element

    Let a be in a group G with |a| = m. If n is relatively prime to m, show that a = b^n for some b in G.

    My Proof so far:

    Now the order of a is m, so I have a^m = e, the identity. n is relatively prime to m, so I have ng + r = m for some integers g and r, with m > r > n.

    umm... what now?
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by tttcomrader View Post
    Let a be in a group G with |a| = m. If n is relatively prime to m, show that a = b^n for some b in G.

    My Proof so far:

    Now the order of a is m, so I have a^m = e, the identity. n is relatively prime to m, so I have ng + r = m for some integers g and r, with m > r > n.

    umm... what now?
    I should think that this would be easier to do if you start from the idea that G contains the subset $\displaystyle \{ e, a, a^2, ~ ... ~, a^{m -1} \}$. Then show that one of these elements must be your "b" for a given n.

    Just a thought.

    -Dan
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  3. #3
    Super Member Rebesques's Avatar
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    n is relatively prime to m, so I have
    Well, everything goes well to that point Now by Euclidean division, there are integers g and r, such that gn+rm=1. So $\displaystyle \alpha^{gn+rm}=\alpha\Rightarrow \alpha^{gn}\alpha^{rm}=\alpha\Rightarrow (\alpha^{g})^n(\alpha^{m})^{r}=\alpha\Rightarrow (\alpha^{g})^{n}=\alpha$, that is $\displaystyle \beta=\alpha^{g}$.





    Ps. Topsq:
    Then show that one of these elements must be your "b" for a given n.
    Suppose $\displaystyle \alpha=\beta^n=\alpha^i$ for some i<n, and prove you chose a bad start
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  4. #4
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    We haven't learn the Euclidean division yet, so I don't think I'm allow to use that. And how to you have the (a^m)^r?

    thanks
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  5. #5
    Super Member Rebesques's Avatar
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    We haven't learn the Euclidean division yet
    Really? Then what is this:

    so I have ng + r = m
    Use that rule and amaze everyone by showing gn+rm=1 for some g,r!


    And how to you have the (a^m)^r?
    a^m=e and e^r=e.
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  6. #6
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    Quote Originally Posted by tttcomrader View Post
    Let a be in a group G with |a| = m. If n is relatively prime to m, show that a = b^n for some b in G.
    Let the order the order of $\displaystyle a$ be $\displaystyle m$. And consider the cyclic subgroup $\displaystyle \left< a\right> = \{a,a^2,...,a^m\}$. Now the order of this cyclic group is $\displaystyle m$. Thus, the generators of this group are $\displaystyle a^k$ where $\displaystyle k$ is relatively prime to $\displaystyle m$. In particular, $\displaystyle k=n$ by hypothesis. That means $\displaystyle \left< a^n \right> = \left< a \right>$. Now $\displaystyle a^n =\{ a^n,a^{2n},...,a^{mn} \}$ is a premutation of $\displaystyle \{ a,a^2, ... ,a^m\}$ thus $\displaystyle a$ is found among $\displaystyle \{a^n,...,a^{nm}\}$ and hence $\displaystyle a=a^{nj} = \left( a^j \right) ^n$. Let $\displaystyle b=a^j$ for some $\displaystyle j$. And the proof is complete.
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