Bogomolov multipliers and retract rationality for semi-direct products

2012/07/25 by Kang, Ming-chang · 2 citations
#14E08 #14L30 #20J06 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1207.5867

Abstract

Let G be a finite group. The Bogomolov multiplier B₀(G) is constructed as an obstruction to the rationality of \bmC(V)^G where G→ GL(V) is a faithful representation over \bmC. We prove that, for any finite groups G₁ and G₂, B₀(G₁× G₂)\xrightarrow∼ B₀(G₁)× B₀(G₂) under the restriction map. If G=N\rtimes G₀ with gcd\|N|,|G₀|\=1, then B₀(G)\xrightarrow∼ 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 (\bmZ/p \bmZ)ⁿ and \bmZ/pⁿ \bmZ respectively. On the other hand, if k is an infinite field and G=N\rtimes G₀ where N is an abelian normal subgroup of exponent e satisfying that ζ_e∈ 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 two previous results of Saltman and Jambor \citeJa.

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Related

• Retract Rational Fields
• Exponents of Bogomolov multipliers
• Retract rationality and Noether's problem
• Frobenius groups and retract rationality
• Noether's Problem on Semidirect Product Groups