Bogomolov multipliers for some p-groups of nilpotency class 2

Abstract

Abstract. 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 θ| K1 : K 1 → 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).

Extending finite free actions of surfaces

“…The HAP package (http://hamilton.nuigalway.ie/Hap/www/), implemented in GAP [31], permits to compute the Bogomolov multiplier of a group 𝐺 of small order (using the command BogomolovMultiplier(𝐺)). The following facts can be found in, for instance, [17,18,21,25].…”

“…The HAP package (http://hamilton.nuigalway.ie/Hap/www/), implemented in GAP [31], permits to compute the Bogomolov multiplier of a group $$G$$ of small order (using the command BogomolovMultiplier($$G$$)). The following facts can be found in, for instance, [17, 18, 21, 25]. (1)$$B_{0}(G)=0$$ if $$G$$ is one of the followings: (a)a symmetric group; (b)a simple group; (c)a $$p$$‐group of order at most $$p^{4}$$; (d)an abelian‐by‐cyclic groups (i.e., $$G$$ contains an abelian group $$A$$ as a normal subgroup such that $$G/A$$ is a cyclic group); (e)a primitive supersolvable group; (f)an extraspecial $$p$$‐group (i.e., the center $$Z_{G}$$ is a cyclic group of order $$p$$ and $$G/Z_{G} \cong {\mathbb {Z}}_{p}^{2n}$$); (2)…”

An exact sequence and triviality of Bogomolov multiplier of groups

“…Rai [18] gave a negative answer to this question by constructing an example. To answer this Question, Michailov [12,Theorem 3.1]…”