The Bogomolov multiplier of rigid finite groups

Abstract

The Bogomolov multiplier 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. This invariant of G plays an important role in birational geometry of quotient spaces V /G. We show that in many cases the vanishing of the Bogomolov multiplier is guaranteed by the rigidity of G in the sense that it has no outer class-preserving automorphisms.

Noether's problem for groups of order 243

“…We remark that it is known that Br v,C (C(G)) = 0 if G is an extraspecial p-group of order p 2n+1 for any prime number p and any positive integer n [25] (also see [16]). But we can show that C(G) is rational only when G is such a group of order 243 (see the following proof) or of order p 3 (by Theorem 1.7).…”

On some closure properties of the non-abelian tensor product

“…The Bogomolov multiplier can be seen as an obstruction to Noether's rationality problem. In the last few years, there has been a lot of research on the class of groups with trivial and non-trivial Bogomolov multiplier (see [5,14,16,12,13]). Except for the Chevalley and Steinberg groups, the Schur multiplier of most of the other finite simple groups have order at most 2.…”

Bogomolov multipliers for some p-groups of nilpotency class 2

“…Kang and Kunyavskiȋ [16] raised the question whether the triviality of B 0 (G 1 ) and B 0 (G 2 ) implies the triviality of B 0 (G). With the aid of (1.1) we prove our first main result, stating that if θ : G 1 → G 2 is a group homomorphism such that its restriction θ| K 1 : 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).…”

“…The reader is referred to the paper [16] for a survey of groups with trivial Bogomolov multipliers. Definition 1.1.…”