Products of conjugacy classes of the infinite symmetric groups (Q793830)

Let \(S_ 0\) be the group of all permutations on a countably infinite set, and let \(p\in S_ 0\) with at least one infinite orbit. Let \(\pi\) be the following property: (\(\pi)\) p has exactly one infinite orbit and for every natural number n at most a finite number of orbits of length n. One of the main results is the following theorem: (a) If \(\pi\) holds then every element in \(S_ 0\backslash(S^ 0_ 0\backslash A_ 0)\) is the product of 2 conjugates of p; (b) if \(\pi\) does \textit{not} hold then every element of \(S_ 0\) is the product of 2 conjugates of p. (In this statement \(S^ 0_ 0\) is the set of permutations in \(S_ 0\) of finite support and \(A_ 0\) is the infinite alternating group in \(S_ 0)\). The other main result involves permutations on sets of arbitrary infinite cardinality, which in the special case of countable cardinality leads to the conjecture: If \(p\in S_ 0\) has infinite support and is not a fixed- point-free involution then every element of \(S_ 0\) is the product of 3 conjugates of p. A remark added in proof states that the author eventually proved that this conjecture is true.