H*(S(co); Z.,) with coefficients in the integers mod p (p :prime). It is proved that the height of any non-zero element is co if p = 2, and is either co or < p if p is odd. If p = 2 this fact and the result in [11, ? 6] enable us to determine the structure of the cohomology algebra H*(S(oo); Z2) by using Borel's theorem on the structure of Hopf algebras. Let S(m) denote the symmetric group of degree m, and S(m, p) a p-Sylow subgroup of S(m). It is also proved that the above result on height is still valid if S(oo) is replaced by S(m) or S(m, p). In fact we derive the result for S(m) and S(oo) from the one for S(m, p). As is well known, S(m, p) is constructed by means of the wreath product. Therefore Part II starts with the consideration of the cohomology of the wreath product. Part III determines the structure of the graded algebra H*(S(cc); Z,) in which the multiplication is given by A* in Part I. It is proved that H*(S(oo); Zp) is isomorphic to the free commutative Z.-algebra generated by a certain set Q(p). Let SPm(Sn) denote the rn-fold symmetric product of an n-sphere, and let u(m) be a generator of the integral cohomology group Hn(SPm(Sn); Z). Assume n is even, and consider the reduced powers u(m)m/a of u(m) with respect to elements a e H,(S(m); G). Then Steenrod states without proof in his lecture notes [14] that if i < n the correspondence a - u(m)m/a yields an isomorphism of H,(S(m); G) onto Hmn-1 (SPm(Sn); G). This fact is basic in the argument of Part III. For the sake of completeness, a proof of the above Steenrod theorem is given in Part IV. The proof is