Let S = {ai}−∞∞ be a
countable set and P any permutation of S with infinite support. Since the subgroup
generated by the conjugacy class 𝒦 of P must be normal in Sym(S), we know that
every permutation of S is a product of permutations from 𝒦 Since it has
recently been discovered that every even permutation in the finite symmetric
group Sym(n) may be expressed as a product of exactly two n-cycles, we
are naturally led to a similar question for Sym (S) and the infinite cycle
C = (⋯,a−2,a−1,a0,a1,a2,⋯), with support all of S. In this paper it is
proved that for each k ≧ 3 every permutation of S is a product of exactly k
cycles conjugate to C, but that no odd finite permutation is a product of
two.
