Bogomolov multipliers for some <Emphasis Type="Italic">p</Emphasis>-groups of nilpotency class 2

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. The triviality of the Bogomolov multiplier is an obstruction to Noether’s problem. We show that if G is a central product of G ₁ and G ₂, regarding K _i ≤ Z(G _i ), i = 1, 2, and θ: G ₁ → G ₂ is a group homomorphism such that its restriction \(\theta {|_{{K_1}}}:{K_1} \to {K_2}\) is an isomorphism, then the triviality of B ₀(G ₁/K ₁),B ₀(G ₁) and B ₀(G ₂) implies the triviality of B ₀(G). We give a positive answer to Noether’s problem for all 2-generator p-groups of nilpotency class 2, and for one series of 4-generator p-groups of nilpotency class 2 (with the usual requirement for the roots of unity).