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Math Help - Disjoint cycles...

  1. #1
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    Disjoint cycles...

    Q1) Write in the following as a single cycle or a product of disjoint cycles?

    A) ( 1 2 3 ) ^(-1) ( 2 4 ) ( 1 2 3 )

    B) ( 1 3 )^(-1) ( 2 4 ) ( 2 3 5 )^(-1)

    I am having some difficulties understanding the difference between the normal set and this inverse sets...I cant figure it out..
    Please help!

    Thanks!
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  2. #2
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    Quote Originally Posted by Vedicmaths View Post
    Q1) Write in the following as a single cycle or a product of disjoint cycles?

    A) ( 1 2 3 ) ^(-1) ( 2 4 ) ( 1 2 3 )
    I do this first one. First (123)=(321)=(132) by the other post. Thus, we need to simplify (132)(24)(123) remember this means \sigma = (132)\circ (24) \circ (123). To do these problems we map 1 and then find \sigma(1) then \sigma(\sigma(1)) until we complete the cycle. Note, 1\to 2 by applying (123) first then 2\to 4 by applying (24) second and 4\to 4 by applying (132) third. Thus, 1\to 4 by \sigma. Now we find \sigma(4). Note 4\to 4 by (123) first then 4\to 2 under (24) and finally 2\to 1 by (132). Thus, \sigma(\sigma(1)) = 1 and it means we only write (14) as we express \sigma as a product of disjoint cycles. Next we do the same with 2. See that 2\to 2 under \sigma and so 3\to 3. Which means the representation is (14)(2)(3) = (14).
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