Bogomolov multipliers for some $p$-groups of nilpotency class 2

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 $K_i\le Z(G_i),i=1,2$, and $\theta :G_1\to G_2$ 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).

This is the revised version which appeared in Acta Math. Sinica (English Series). arXiv admin note: text overlap with arXiv:1304.1890

References in corpus (1)

• Noether's problem for abelian extensions of cyclic $p$-groups

Cited by in corpus (2)

• Bogomolov multipliers of some groups of order $p^6$
• Bogomolov multiplier and the Lazard correspondence