Permutations as products of conjugate infinite cycles
Edward A. Bertram · Pacific Journal of Mathematics · 1971
Let S = {aiiZoo be a countable set and P any permutation of S with infinite support.Since the subgroup generated by the conjugacy class 3ίΓ of P must be normal in Sym (S), we know that every permutation of 5 is a product of permutations from 3ίΓ.Since it has recently been discovered that every even permutation in the finite symmetric group Sym in) 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~l t α 0 , a ίf a 2 , •), with support all of S. In this paper it is proved that for each k Ξ> 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.