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18.49 (2014)

Solved

Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying $1 < a, b, c \leqslant n - 2$ the symmetric group $S_n$ has elements of order $a$ and $b$ whose product has order $c$?

Contributor: S. Kohl

Progress

Yes, it is (G. A. Miller, Amer. J. Math., 22, no. 2 (1900), 185–190). Another solution is in (J. König, Eur. J. Comb., 57 (2016), 50–56).

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