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Noether’s problem and unramified Brauer groups. (English) Zbl 1291.13012
Let \(K\) be a field and \(G\) be a finite group. Let \(G\) act on the rational function field \(K(x(g):g\in G)\) by \(K\) automorphisms defined by \(g\cdot x(h)=x(gh)\) for any \(g,h\in G\). Denote by \(K(G)\) the fixed field \(K(x(g):g\in G)^G\). Noether’s problem then asks whether \(K(G)\) is rational (i.e., purely transcendental) over \(K\). The Bogomolov multiplier \(B_0(G)\) of a finite group \(G\) is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of \(G\). A necessary condition for the positive answer of Noether’s problem is that the Bogomolov multiplier \(B_0(G)\) is trivial. The authors study the Bogomolov multiplier of the groups of order \(p^5\) for an odd prime \(p\). They show that \(B_0(G)\neq 0\) if and only if \(G\) belongs to the isoclinism family \(\Phi_{10}\) in James’ classification of groups of order \(p^5\) R. James [Math. Comput. 34, 613–637 (1980; Zbl 0428.20013)]. The triviality of the Bogomolov multiplier for \(G\notin \Phi_{10}\) was obtained independently with other methods by P. Moravec [J. Algebra 372, 320–327 (2012; Zbl 1303.13010)].

MSC:
13A50 Actions of groups on commutative rings; invariant theory
14E08 Rationality questions in algebraic geometry
14M20 Rational and unirational varieties
20J06 Cohomology of groups
12F12 Inverse Galois theory
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