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Math Help - Permutations & Cycles

  1. #1
    Super Member craig's Avatar
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    Permutations & Cycles

    In S_6, let \beta = (61)(65)(153)(653)

    I found the cycle decomposition of \beta to be (1)(2)(356)(4), the question then asks me what is the cycle length of \beta.

    This is probably a stupid question but is it cycle length 4? I think I missed that small part of the lecture...

    Thanks in advance
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  2. #2
    Member Haven's Avatar
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    In Cycle notation, 1-cycles are often omitted, so you can right
    \beta = (1)(2)(356)(4) = (356)
    So, we are left with a cycle of length 3 (since it permutes 3 numbers)

    Now if we want to determine if this is an even or odd permutation, we must decompose \beta into transpositions (2-cycles)

    so,  \beta = (36)(35)
    Since there are 2 transpositions, \beta is an even permutation
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  3. #3
    Super Member craig's Avatar
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    Quote Originally Posted by Haven View Post
    In Cycle notation, 1-cycles are often omitted, so you can right
    \beta = (1)(2)(356)(4) = (356)
    So, we are left with a cycle of length 3 (since it permutes 3 numbers)

    Now if we want to determine if this is an even or odd permutation, we must decompose \beta into transpositions (2-cycles)

    so,  \beta = (36)(35)
    Since there are 2 transpositions, \beta is an even permutation
    Of course, 1 cycles just send the number to itself. Thanks for the reply, appreciate it.
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