Bogomolov multipliers and retract rationality for semidirect products
Journal
Journal of Algebra
Journal Volume
397
Pages
407-425
Date Issued
2014
Author(s)
Abstract
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 → G L(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]. © 2013 Elsevier Inc.
Type
journal article