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In a solution to a question, there is a line that reads in the group $S_{10}$: $$(1,8,2,5)^2 = (1,2,8,5)$$.

It is surely not meant to read: $$(1,8,2,5)^2 = (1,2)(5,8)$$ or am I mistaken about how cycles work?

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    $\begingroup$ No, you're right. It must be a typo. $\endgroup$ – Bernard May 12 at 17:17
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Indeed, $(1825)^2 = (12)(58)$, and the solution has a typo. You can check that it is a typo quickly by seeing that as $4$-cycles have order $4$, squaring any $4$-cycle must give an element of order $2$, and so one can never obtain a $4$-cycle by squaring a $4$-cycle.

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