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

Leary, I. J.

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)

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

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