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The mod-$$p$$ cohomology rings of some $$p$$-groups. (English) Zbl 0782.20042
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)

MSC:
 20J06 Cohomology of groups 20F05 Generators, relations, and presentations of groups 20D15 Finite nilpotent groups, $$p$$-groups 55T10 Serre spectral sequences
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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
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