The symmetric group on an infinite set does not have a subgroup of index 2, as Vitali (1915) proved that each permutation can be written as a product of three squares. (Any squared element must belong to the hypothesized subgroup of index 2, hence so must the product of any number of squares.) However it contains the normal subgroup S of permutations that fix all but finitely many elements, which is generated by transpositions. Those elements of S that are products of an even number of transpositions form a subgroup of index 2 in S, called the alternating subgroup A. Since A is even a characteristic subgroup of S, it is also a normal subgroup of the full symmetric group of the infinite set. The groups A and S are the only nontrivial proper normal subgroups of the symmetric group on a countably infinite set. This was first proved by Onofri (1929) and independently Schreier – Ulam (1934). For more details see (Scott 1987, Ch. 11.3) or (Dixon & Mortimer 1996, Ch. 8.1).