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Suppose you have a permutation $\sigma = \sigma_1 \sigma_2$ where $\sigma_1 = (i_1 i_2...i_k)$, $\sigma_2 = (j_1 j_2...j_l)$ and $i_{k_1},i_{k_2},...,i_{k_r}$ are equal to $j_{l_1},j_{l_2},...,j_{l_r}$ respectively (in other words the cycles are non disjoint). Is there any way to represent $\sigma$ as a single cycle?

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    $\begingroup$ it need not be representable as a "single" cycle all the time. For example $\alpha=(1 \, 2 \,3)$ and $\beta=(1 \, 2 \, 4)$ will give $\alpha \beta=(1 \, 3)(2 \, 4)$. $\endgroup$ – Anurag A May 24 '15 at 20:07
  • $\begingroup$ Thanks. I did not realize that. $\endgroup$ – Iguana May 24 '15 at 20:20
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Not in general, no. For example, $$(1234)(1234)=(13)(24)$$ which is not a cycle.

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