Bogomolov multipliers for some pp-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 $K_i\leq Z(G_i), i=1,2$, and $\theta:G_1\to G_2$ is a group homomorphism such that its restriction $\theta\vert_{K_1}:K_1\to K_2$ 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).Comment: This is the revised version which appeared in Acta Math. Sinica (English Series). arXiv admin note: text overlap with arXiv:1304.189

On realizability of pp-groups as Galois groups

In this article we survey and examine the realizability of $p$-groups as Galois groups over arbitrary fields. In particular we consider various cohomological criteria that lead to necessary and sufficient conditions for the realizability of such a group as a Galois group, the embedding problem (i.e., realizability over a given subextension), descriptions of such extensions, automatic realizations among $p$-groups, and related topics.Comment: In this version is added an example at the end of Section 6. Also, some mistakes are corrected and the references are update

Induced orthogonal representations of Galois groups

We prove a result showing the connection between induced orthogonal representations and corestrictions of group extensions derived from their Clifford groups. By the means of the corestriction map we then obtain new obstructions to the μ p -embedding problems given by the group extensions of the modular p-group, and to one μ 2 -embedding problem given by a group extension of the dihedral group

Noether's Problem for p-Groups with an Abelian Subgroup of Index p

Let K be a field and G be a finite group. Let G act on the rational function field K(x(g) : g ∈ G) by K-automorphisms defined by g · x(h) = x(gh) for any g, h ∈ G. Denote by K(G) the fixed field K(x(g) : g ∈ G)G. Noether's problem then asks whether K(G) is rational over K. Let p be an odd prime and let G be a p-group of exponent pe. Assume also that (i) char K = p &gt; 0, or (ii) char K ≠ p and K contains a primitive pe-th root of unity. In this paper we prove that K(G) is rational over K for the following two types of groups: (1) G is a finite p-group with an abelian normal subgroup H of index p such that H is a direct product of normal subgroups of G of the type Cpb × (Cp)c for some b, c with 1 ≤ b and 0 ≤ c; (2) G is any group of order p5 from the isoclinic families with numbers 1, 2, 3, 4, 8 and 9. </jats:p

Съвременна теория на Галоа и нейните класически задачи

[Michailov Ivo M.; Михайлов Иво М.]In this survey we outline the milestones of several classical problems in Galois theory: Noether’s problem, the inverse problem and the embedding problem. We demonstrate the connection between these problems in the bigger context of the invariant theory of finite groups. We also point out several new results of the author and his collaborators regarding Noether’s problem and the embedding problem. 2020 Mathematics Subject Classification: 12F12, 13A50, 20D15, 14E08

On Galois cohomology and realizability of 2-groups . . .

In this paper we develop some new theoretical criteria for the realizability of -groups as Galois groups over arbitrary fields. We provide necessary and sufficient conditions for the realizability of 14 of the 22 non-abelian 2-groups having a cyclic subgroup of index 4 that are not direct products of groups