Leary, I. J. The mod-\(p\) cohomology rings of some \(p\)-groups. (English) Zbl 0782.20042 Math. Proc. Camb. Philos. Soc. 112, No. 1, 63-75 (1992). Let \(p\) be an odd prime and \[ P(n) = \langle A,B,C\mid A^ p = B^ p = C^{p^{n-2}} = [A,C] = [B,C] = 1,\;[A,B] = C^{p^{n-3}}\rangle \] where \(n \geq 3\). The groups \(P(n)\) have order \(p^ n\) and are central extensions of a cyclic subgroup by \(C_ p \oplus C_ p\). In this very interesting paper the author studies the mod-\(p\) cohomology ring of \(P(n)\) and calculates the mod-\(p\) cohomology ring of \(P(3)\). This completes the calculation of the mod-\(p\) cohomology rings of the groups of order \(p^ 3\). The other cases have been done by Pham Anh Minh and Huynh Mui [in Acta Math. Vietnam 7, 17-26 (1982; Zbl 0596.20046)] and T. Diethelm [in Arch. Math. 44, 29-38 (1985; Zbl 0548.20040)]. R. Milgram and M. Tezuka [in The geometry and cohomology of \(M_{12}\). II] have shown that the mod-3 cohomology of \(P(3)\) is detected by the restrictions to proper subgroups, however this is not so for \(p \geq 5\). There is some overlap between this paper and results of D. J. Benson and J. F. Carlson [in Bull. Lond. Math. Soc. 24, 209-235 (1992)]. The author’s method involves embedding \(P(n)\) in the non-abelian Lie group whose underlying topological space consists if \(p^ 2\) circles. He used a similar method for his calculation of the integral cohomology of the same groups [in Math. Proc. Camb. Philos. Soc. 110, 25-32 (1991; Zbl 0736.20026)]. Reviewer: O.Talelli (Athens) Cited in 1 ReviewCited in 24 Documents MSC: 20J06 Cohomology of groups 20F05 Generators, relations, and presentations of groups 20D15 Finite nilpotent groups, \(p\)-groups 55T10 Serre spectral sequences Keywords:central extensions; mod-\(p\) cohomology rings; restrictions; integral cohomology PDF BibTeX XML Cite \textit{I. J. Leary}, Math. Proc. Camb. Philos. Soc. 112, No. 1, 63--75 (1992; Zbl 0782.20042) Full Text: DOI arXiv References: [1] Burnside, Theory of Finite Groups (1897) · JFM 01.0191.01 [2] Brown, Cohomology of Groups (1982) · doi:10.1007/978-1-4684-9327-6 [3] Araki, Mem. Fac. Sci. Kyusyu Univ. Ser. A 11 pp 1– (1957) [4] Vasquez, Bol. Soc. Math. Mexicana 2 pp 1– (1957) [5] Minh, Acta Math. Vietnam. 7 pp 17– (1982) [6] Diethelm, Arch. Math. (Basel) 44 pp 29– (1985) · Zbl 0548.20040 · doi:10.1007/BF01193778 [7] DOI: 10.2307/1994856 · Zbl 0217.34903 · doi:10.2307/1994856 [8] DOI: 10.2307/1994385 · Zbl 0146.19201 · doi:10.2307/1994385 [9] Leary, Math. Proc. Cambridge Philos. Soc. 110 pp 25– (1991) [10] DOI: 10.1016/0021-8693(89)90311-6 · Zbl 0696.55025 · doi:10.1016/0021-8693(89)90311-6 [11] DOI: 10.1016/0021-8693(89)90310-4 · Zbl 0696.55024 · doi:10.1016/0021-8693(89)90310-4 [12] DOI: 10.1016/0021-8693(69)90027-1 · Zbl 0192.34302 · doi:10.1016/0021-8693(69)90027-1 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.