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    Does anyone know of a website with a good method for proving that any permutation r of {1,..,n} can be expressed as a product of disjoint cycles?
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    (Original post by flyinghorse)
    Does anyone know of a website with a good method for proving that any permutation r of {1,..,n} can be expressed as a product of disjoint cycles?
    It's a fairly simple inductive proof anyway.

    Take 1 and consider the cycle

    (1 r(1) r^2(1) ...)

    For some k, r^k(1)=1. The reason why this must happen is that the list

    1,r(1), ..., r^i(1),...

    can't all be distinct (only finitely many elements) and so r^i(1) = r^j(1) for i<j and then r^(j-i)(1)=1.

    So this cycle has length k. Either k=n and you're done, or at some point (k<n) the cycle closes with r^k(1)=1. But in either case you've removed some k of the n elements, and r restricts to a permutation of the remaining n-k elements. By an inductive hypothesis this perm of n-k elements can be written as a product of disjoint cycles.
 
 
 

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