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Math Help - cycle decomposition 3

  1. #1
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    cycle decomposition 3

    Express as the product of disjoint cycles and find the order.

    a) (1 2 3 5 7)(2 4 7 6)
    b) (1 2)(1 3)(1 4)
    e) (1 2 3)(3 5 7 9)(1 2 3)^-1
    f) (1 2 3 4 5)^3.

    Please show steps. Thanks!
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  2. #2
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    Quote Originally Posted by mpryal View Post
    Express as the product of disjoint cycles and find the order.

    a) (1 2 3 5 7)(2 4 7 6)
    If a permutation is expressed as a product of disjoint cycles then the order is the lowest common multiple of all the cycle lengths. This permutation is not a product of disjoint cycles, so we have to turn it into a product of disjoint cycles.

    Note (under this permutation):
    1\mapsto 2\mapsto 4\mapsto 1 \mapsto ...
    3\mapsto 5 \mapsto 7 \mapsto 6 \mapsto 3 \mapsto...

    Thus, (12357)(2476) = (124)(3576)
    This means the order is \text{lcm}(3,4)=12

    Try the other ones and show your work.
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