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Thread: Abelian subgroups of S_{n}

  1. #1
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    Abelian subgroups of S_{n}

    I am finding all abelian subgroups of a symmetric group $\displaystyle S_{n}$ for n>=5.

    What I have found so far is
    1. {e} : trivial group
    2. a cyclic group of order n
    3. a quotient group of order 2: $\displaystyle S_{n}/A_{n}$.

    I am wondering if below 4 is correct.
    4. "a cyclic group of order k which is a divisor of n"

    Any more subgroup exists for $\displaystyle S_{n}$?
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  2. #2
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    Quote Originally Posted by aliceinwonderland View Post

    I am finding all abelian subgroups of a symmetric group $\displaystyle S_{n}$ for n>=5.
    i don't think this is basically possible to do because we even don't know all cyclic subgroups of $\displaystyle S_n.$ no formula is known for the maximum order of elemens of $\displaystyle S_n.$ however, there's a result due

    to Landau that if $\displaystyle f(n)$ is the maximum order of elements of $\displaystyle S_n,$ then $\displaystyle \lim_{n\to\infty} \frac{\ln f(n)}{\sqrt{n \ln n}} = 1.$ it's not surprising that there's no closed form for $\displaystyle f(n),$ because $\displaystyle f(n)$ is related to the partition function.
    Last edited by NonCommAlg; Jan 6th 2009 at 09:08 PM. Reason: a typo!
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    i don't think this is basically possible to do because we even don't know all cyclic subgroups of $\displaystyle S_n.$ no formula is known for the maximum order of elemens of $\displaystyle S_n.$
    Any concrete example of an element of $\displaystyle S_n$ that has an order bigger than n? It is easy to find an element of order n in $\displaystyle S_n$, but it is hard for me to find an element that has an order bigger than n.
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  4. #4
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    Quote Originally Posted by aliceinwonderland View Post
    Any concrete example of an element of $\displaystyle S_n$ that has an order bigger than n? It is easy to find an element of order n in $\displaystyle S_n$, but it is hard for me to find an element that has an order bigger than n.
    Sure, take $\displaystyle S_7$ and consider $\displaystyle (1234)(567)$.
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