On the group of permutations with countable support
Justin T. Lloyd, William H. Smiley · Pacific Journal of Mathematics · 1975
Let S x denote the group of permutations of the set X.If H α 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 5χ generate the subgroup of permutations with countable support.If /E5 X and x EX, then let O f (x) denote the orbit of x under /.The set X is the disjoint union of the distinct orbits of / [1].In case f(x)^x, O f (x) is called a nontriυial orbit of /.Let S(f) denote the support of the permutation /.If S(f) consists of exactly one nontrivial "orbit, then / is called a cycle.Let C x be the subgroup of S x consisting of all finite products of cycles.If X is finite, then C x = S x .For an uncountable set X, C x is a proper subgroup of S x .We now show that C x = S x in the remaining case.THEOREM.// X is countably infinite, then S x is generated by its cycles.