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Math Help - cycle decomposition 1

  1. #1
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    cycle decomposition 1

    Show that if f,g are 2 disjoint cycles, then fg = gf.

    Please show steps. Thanks!
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  2. #2
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    Quote Originally Posted by mpryal View Post
    Show that if f,g are 2 disjoint cycles, then fg = gf.

    Please show steps. Thanks!
    Let f be the cycle (a_1,...,a_i) and g be the cycle (b_1,...,b_j) where a_1,...,a_i,b_1,....,b_j are distinct. To show that fg =  gf you need to show fg(x) = gf(x) for any x. If x is distinct from a_1,...,a_i,b_1,....,b_j then fg(x) = x = fg(x). If x\in \{a_1,...,a_i\}, then g(x) = x while g(f(x)) = f(x) since f(x) \in \{a_1,...,a_i\} so fg(x) = f(x) = g(f(x)) = gf(x). The same idea is when x\in \{b_1,...,b_j\} and so we have that fg=gf.
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