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Math Help - 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
    Super Member 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 n can be considered as the set of all permutations of n letters.

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

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

    Thus:
    <br />
\rho \tau \rho^{-1} (i) = \rho \tau  (i) = \rho (j) = k<br />

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

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

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

    So no non-identity elements of S_n commute with all elements of S_n.
    Last edited by Matt Westwood; July 4th 2008 at 12:43 PM. 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 \tau(i)?
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  5. #5
    Super Member Matt Westwood's Avatar
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    In this context, S_n is the set of all permutations of n objects (which we can call 1, 2, 3, ..., n without loss of generality).

    A permutation of 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.

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

    So \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 \tau is the permutation that changes (1, 2, 3) to (1, 3, 2), then \tau (1) = 1, \tau (2) = 3 and \tau (3) = 2.

    In this context, \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 i.

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

    Don't confuse S_n (the set of permutations on n obects) with the actual objects that 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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