Bogomolov multipliers and retract rationality for semidirect products

• Published in: Journal of algebra 2014-01, Vol.397, p.407-425
• Main Author: Kang, Ming-chang
• Format: Article
• Language: English
• Subjects: Bogomolov multiplier; Cohomology of finite groups; Noetherʼs problem; Rationality problem; Retract rationality; Unramified Brauer groups
• ISSN: 0021-8693; 1090-266X
• DOI: 10.1016/j.jalgebra.2013.08.039

Let G be a finite group. The Bogomolov multiplier B0(G) is constructed as an obstruction to the rationality of C(V)G where G→GL(V) is a faithful representation over C. We prove that, for any finite groups G1 and G2, B0(G1×G2)→∼B0(G1)×B0(G2) under the restriction map. If G=N⋊G0 with gcd{|N|,|G0|}=1, then B0(G)→∼B0(N)G0×B0(G0) under the restriction map. For any integer n, we show that there are non-direct product p-groups G1 and G2 such that B0(G1) and B0(G2) contain subgroups isomorphic to (Z/pZ)n and Z/pnZ respectively. On the other hand, if k is an infinite field and G=N⋊G0 where N is an abelian normal subgroup of exponent e satisfying that ζe∈k, we will prove that, if k(G0) is retract k-rational, then k(G) is also retract k-rational provided that certain “local” conditions are satisfied; this result generalizes previous results of Saltman and Jambor [18].