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 or . Maximal subgroups The maximal subgroups of Sn fall into three classes: the intransitive, the imprimitive, and the primitive. The intransitive maximal subgroups are exactly those of the form for . The imprimitive maximal subgroups are exactly those of the form , where is a proper divisor of n and "wr" denotes the wreath product. The primitive maximal subgroups are more difficult to identify, but with the assistance of the O'Nan–Scott theorem and the classification of finite simple groups, gave a fairly satisfactory description of the maximal subgroups of this type, according to . Sylow subgroups The Sylow subgroups of the symmetric groups are important examples of p-groups. They are more easily described in special cases first: The Sylow p-subgroups of the symmetric group of degree p are just the cyclic subgroups generated by p-cycles. There are such subgroups simply by counting generators. The normalizer therefore has order and is known as a Frobenius group (especially for ), and is the affine general linear group, . The Sylow p-subgroups of the symmetric group of degree p2 are the wreath product of two cyclic groups of order p. For instance, when , a Sylow 3-subgroup of Sym(9) is generated by and the elements , and every element of the Sylow 3-subgroup has the form for . The Sylow p-subgroups of the symmetric group of degree pn are sometimes denoted Wp(n), and using this notation one has that is the wreath product of Wp(n) and Wp(1). In general, the Sylow p-subgroups of the symmetric group of degree n are a direct product of ai copies of Wp(i), where and (the base p expansion of n). For instance, , the dihedral group of order 8, and so a Sylow 2-subgroup of the symmetric group of degree 7 is generated by and is isomorphic to . These calculations are attributed to and described in more detail in . Note however that attributes the result to an 1844 work of Cauchy, and mentions that it is even covered in textbook form in . Transitive subgroups A transitive subgroup of Sn is a subgroup whose action on {1, 2, ,..., n} is transitive. For example, the Galois group of a (finite) Galois extension is