If we will look at the cycle structure of $S_4$, then we have $6$ different $4$-cycles. They are $$(1234), (1243), (1324), (1342), (1423), (1432).$$ And we have $3$ disjoint multiplying transpositions. They are $$(12)(34), (13)(24), (14)(23).$$
I can determine them by thinking of partition of $4$.
They have relation as follows: $$(12)(34)=(1324)^2=(1423)^2$$ $$(13)(24)=(1234)^2=(1432)^2$$ $$(14)(23)=(1243)^2=(1342)^2$$
- What does this relation tell us?
- I wonder this: Can we find $n$-cycle permutaton in $S_n$ such that any power of it is a product of all different disjoint transposition? For example; Does there exist any $10$- cycle $\alpha$ in $S_{10}$ such that $\alpha^k=(1\ 2)\ (3\ 4)\ (5\ 6)\ (7\ 8)\ (9\ 10)$.
$\textbf{My attempt:}$ Let $\beta= (1\ 2)\ (3\ 4)\ (5\ 6)\ (7\ 8)\ (9\ 10)$. Then $\beta^2=1$. Assuming there exist any $\alpha $ for suitable $k$ we have $(\alpha^k)^2=\beta^2=1\Longrightarrow 2k\mid 10\Longrightarrow k\mid 5.$ Let $k=5$. Then $\alpha^5=\beta$. I could not continue after that. I guess, we have similar relation as $S_4$.
- I don't want to find $\alpha$. I want to say about something existence of it.