The integral cohomology rings of groups of order $p\sp{3}$

Gene Lewis
1968 Transactions of the American Mathematical Society  
0. Introduction. Most of the papers in cohomology of groups have been theoretical [1], [3]-[6], [8], [9], [13], [14], rather than computational. The following list describes all the (published and unpublished) computations I am aware of. Atiyah [1, p. 60] calculated the integral (cohomology) ring of the quaternion group Q of order 8. Wall [16] obtained the additive structure of split cyclic-by-cyclic extension. He could not get the ring structure. Cartan-Eilenberg [3, Chapter XII, §7]
more » ... the cohomology of a cyclic group with any coefficients, and gave an explicit diagonal map. Evens [6] wrote down the cohomology ring of the dihedral group D of order 8. Unfortunately, he did not publish the details of his calculations. Nakaoka [9] calculated the homology ring of symmetric groups with coefficients Zp and in [10, p. 52] found the cohomology ring of S4, coefficients Z2. Cardenas [2] calculated the ring of 2p2, coefficients Zp. Täte [15] got the integral homology ring of any finitely generated abelian group. Finally, Norman Hamilton and Arnold Shapiro got the integral cohomology ring of Zp I Zp, but neglected to write it down. In this paper, I will find the integral cohomology rings of groups of order /»3, except for Q and D. I merely sketch the abelian cases, as they are trivial.
doi:10.1090/s0002-9947-1968-0223430-6 fatcat:sn6tvui5wndn7dkyz26thxzvba