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Thread: Group Theory

  1. #1
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    Group Theory

    a) If a group has order 12 , then G contains an element of order 6.

    b) If a group has order 18, then G contains an element of order 3.

    If it's true, prove it; if it's false find a counter example.

    I know I should know this but for some reason can't get my head around it. Can someone give me a push in the right direction. Do I have to include the Sylow Theorems?

    Thanx
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  2. #2
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    Quote Originally Posted by bex23 View Post
    a) If a group has order 12 , then G contains an element of order 6.
    This is not true for $\displaystyle A_4$ look here for the details.

    b) If a group has order 18, then G contains an element of order 3.
    Since $\displaystyle 3|18$ by Cauchy's theorem there is an element of order 3.
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  3. #3
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    [quote=ThePerfectHacker;177440]This is not true for $\displaystyle A_4$ look here for the details.

    What about the group D_6. Doesn't that have 2 elements of order 6.
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  4. #4
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    Quote Originally Posted by bex23 View Post
    What about the group D_6. Doesn't that have 2 elements of order 6.
    That is true. Because $\displaystyle D_6$ is generated by $\displaystyle a=(123456)$ and $\displaystyle b=(26)(35)$. Thus. $\displaystyle D_6 = \{ a^ib^j | 0\leq i\leq 5, 0\leq j\leq 1\}$.
    The elements $\displaystyle a^i b$ are all reflections and they have order 2.
    While the order of $\displaystyle a^i$ is $\displaystyle 6/\gcd(i,6)$.
    To have order 6 we require $\displaystyle \gcd(i,6)=1$ thus $\displaystyle i=1,5$.
    This means only $\displaystyle a$ and $\displaystyle a^5$ have order $\displaystyle 6$.

    But what is the problem? The above example is when it is true. But the example with $\displaystyle A_4$ is when it is false. Thus, this statement cannot be true i.e. order 12 subgroups has elements of order 6.
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