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Thread: centralizer of sym(n)

  1. #1
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    centralizer of sym(n)

    Can u help me to prove that the Z(Sym(n))={1}, centralizer of the symmetric group is equal to 1.
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  2. #2
    MHF Contributor Matt Westwood's Avatar
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    Centraliser of symmetric group order > 3 is trivial

    The proof I've got starts like this ...

    The symmetric group order $\displaystyle n$ can be considered as the set of all permutations of $\displaystyle n$ letters.

    Let us choose a permutation $\displaystyle \tau \in Z (S_n), \tau \neq e$, so that $\displaystyle \tau (i) = j, i \neq j$.

    Since $\displaystyle n \ge 3$, we can find another permutation $\displaystyle \rho$ which interchanges j and k $\displaystyle (k \ne i, j)$ and leaves everything else where it is. So the inverse of $\displaystyle \rho$ does the same thing, and both $\displaystyle \rho$ and $\displaystyle \rho^{-1}$ leave i alone.

    Thus:
    $\displaystyle
    \rho \tau \rho^{-1} (i) = \rho \tau (i) = \rho (j) = k
    $

    So $\displaystyle \rho \tau \rho^{-1} (i) = k \ne j = \tau (i)$.

    If $\displaystyle \rho$ and $\displaystyle \tau $ were to commute, $\displaystyle \rho \tau \rho^{-1} (i) = \tau (i)$ (right multiply by $\displaystyle \rho$). But they don't.

    Whatever $\displaystyle \tau \in S_n$ is, you can always find a $\displaystyle \rho$ such that $\displaystyle \rho \tau \rho^{-1} \ne \tau $.

    So no non-identity elements of $\displaystyle S_n$ commute with all elements of $\displaystyle S_n$.
    Last edited by Matt Westwood; Jul 4th 2008 at 11:43 AM. Reason: Correcting syntax errors
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  3. #3
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    centralizer of Sym(n)

    Thanx.
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  4. #4
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    centralizer of Sym(n)

    what do u mean by $\displaystyle \tau(i)$?
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  5. #5
    MHF Contributor Matt Westwood's Avatar
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    In this context, $\displaystyle S_n$ is the set of all permutations of $\displaystyle n$ objects (which we can call 1, 2, 3, ..., $\displaystyle n$ without loss of generality).

    A permutation of $\displaystyle n$ objects is a rearrangement of them. For example, a rearrangement of (1, 2, 3) is (1, 3, 2), and another one is (2, 1, 3), and so on.

    $\displaystyle \tau$ is an element of $\displaystyle S_n$, and so it is a rearrangement of $\displaystyle n$ objects.

    So $\displaystyle \tau$ applied to one of the objects of an n-element set will either move it to somewhere else or leave it where it is.

    For example, if $\displaystyle \tau$ is the permutation that changes (1, 2, 3) to (1, 3, 2), then $\displaystyle \tau (1) = 1, \tau (2) = 3$ and $\displaystyle \tau (3) = 2$.

    In this context, $\displaystyle \tau$ is any permutation you care to think of, but not the identity permutation (which leaves everything where it is). So there's bound to be at least one element of this n-element set which is not left where it is.

    Call this element $\displaystyle i$.

    What I mean by $\displaystyle \tau(i)$ is the result of applying this arbitrarily selected permutation $\displaystyle \tau$ to an equally arbitrarily selected element $\displaystyle i$, and calling it $\displaystyle j$.

    Don't confuse $\displaystyle S_n$ (the set of permutations on $\displaystyle n$ obects) with the actual objects that $\displaystyle S_n$ is acting on. They are not the same thing.
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  6. #6
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    centralizer of Sym(n)

    thanx matt.
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