Bogomolov multipliers and retract rationality for semidirect products

Let G   be a finite group. The Bogomolov multiplier B0(G)B0(G) is constructed as an obstruction to the rationality of C(V)GC(V)G where G→GL(V)G→GL(V) is a faithful representation over CC. We prove that, for any finite groups G1G1 and G2G2, B0(G1×G2)→∼B0(G1)×B0(G2) under the restriction map. If G=N⋊G0G=N⋊G0 with gcd{|N|,|G0|}=1gcd{|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 G1G1 and G2G2 such that B0(G1)B0(G1) and B0(G2)B0(G2) contain subgroups isomorphic to (Z/pZ)n(Z/pZ)n and Z/pnZZ/pnZ respectively. On the other hand, if k   is an infinite field and G=N⋊G0G=N⋊G0 where N is an abelian normal subgroup of exponent e   satisfying that ζe∈kζe∈k, we will prove that, if k(G0)k(G0) is retract k  -rational, then k(G)k(G) is also retract k-rational provided that certain “local” conditions are satisfied; this result generalizes previous results of Saltman and Jambor [18].