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Math Help - Permutation Groups

  1. #1
    Newbie
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    Aug 2010
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    San Antonio, Sibulan, Negros Oriental, Philippines
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    10

    Post Permutation Groups

    Can you please show me the proof of this theorem?
    " Every permutation under Sn where n>1 is a product of two cycles"
    thanks
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  2. #2
    Member
    Joined
    Apr 2008
    Posts
    191
    Clearly the identity can be expressed as (12)(12) and it is a product of 2-cycles. And since you know that every permutation of a finite set can be written as a cycle or as a product of disjoint cycles (proof?), every permutation can be expressed in the form:

    (a_1 a_2...a_k)(b_1b_2...b_t)...(c_1c_2...c_s)

    then you will have:

     <br />
(a_1a_k)(a_1a_{k-1})...(a_1a_2)(b_1b_t)(b_1b_{t-1})...(b_1b_2)...(c_1c_s)(c_1c_{s-1})...(c_1c_2)<br />
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