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Math Help - cycle decomposition 5

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    cycle decomposition 5

    Show that a k-cycle has order k.

    Please show steps. Thanks!
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  2. #2
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    Quote Originally Posted by mpryal View Post
    Show that a k-cycle has order k.

    Please show steps. Thanks!
    Let f=(0,1,2,...,k-1) be the cycle. Then f(n) = n if n\not \in \{0,1,2,...,k-1\} and f(m) = m+1(\bmod k) if m\in \{0,1,2,...,k-1\}. It is seems from this that f^j (n) = n and f^j(m) = m + j(\bmod k). For f^j to be the identity we want f^j (m) = m, so m+j(\bmod k) = m. This happens if and only if k|j. The smallest such j is therefore k=j.
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