On the group of permutations with countable support

Let S x denote the group of permutations of the set X. If H is an infinite cardinal, the set of permutations having support with cardinality less than or equal to H a is a normal subgroup of S x . The principal result of this paper is a constructive proof that S x is generated by its cycles, if X is countably infinite. Of particular interest is the corollary that for any set X, the cycles of 5 generate the subgroup of permutations with countable support.