arXiv subjects

Ivo M. Michailov

Publications and source records attributed to Ivo M. Michailov.

8 recordsLinked to original sources

arXiv2016

Bogomolov multipliers for some $p$-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 $θ:G_1\to G_2$ is a group homomorphism such that its restriction $θ\vert_{K_1}:K_1\to K_2$ 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).

math.GR↗

arXiv2013

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

Let $K$ be a field and $G$ be a finite group. Let $G$ act on the rational function field $K(x(g):g\in G)$ by $K$ automorphisms defined by $g\cdot x(h)=x(gh)$ for any $g,h\in G$. Denote by $K(G)$ the fixed field $K(x(g):g\in G)^G$. Noether's problem then asks whether $K(G)$ is rational (i.e., purely transcendental) over $K$. Let $p$ be any prime and let $G$ be a $p$-group of exponent $p^e$. Assume also that {\rm (i)} char $K = p>0$, or {\rm (ii)} char $K \ne p$ and $K$ contains a primitive $p^e$-th root of unity. In this paper we prove that if $G$ is any $p$-group of nilpotency class 2, which has the ABC (Abelian-By-Cyclic) property, then $K(G)$ is rational over $K$. We also prove the rationality of $K(G)$ over $K$ for two 3-generator $p$-groups $G$ of arbitrary nilpotency class.

math.AG↗

arXiv2013

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 $K(x(g):g\in G)$ by $K$-automorphisms defined by $g\cdot x(h)=x(gh)$ for any $g,h\in G$. Denote by $K(G)$ the fixed field $K(x(g):g\in G)^G$. 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 $p^e$. Assume also that {\rm (i)} char $K = p>0$, or {\rm (ii)} char $K \ne p$ and $K$ contains a primitive $p^e$-th root of unity. In this paper we prove that $K(G)$ is rational over $K$ for the following two types of groups: {\rm (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 $C_{p^b}\times (C_p)^c$ for some $b,c:1\leq b,0\leq c$; {\rm (2)} $G$ is any group of order $p^5$ from the isoclinic families with numbers $1,2,3,4,8$ and 9.

math.AG↗

arXiv2013

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

Let $K$ be a field and $G$ be a finite group. Let $G$ act on the rational function field $K(x(g):g\in G)$ by $K$ automorphisms defined by $g\cdot x(h)=x(gh)$ for any $g,h\in G$. Denote by $K(G)$ the fixed field $K(x(g):g\in G)^G$. Noether's problem then asks whether $K(G)$ is rational (i.e., purely transcendental) over $K$. The first main result of this article is that $K(G)$ is rational over $K$ for a certain class of $p$-groups having an abelian subgoup of index $p$. The second main result is that $K(G)$ is rational over $K$ for any group of order $p^5$ or $p^6$ ($p$ is an odd prime) having an abelian normal subgroup such that its quotient group is cyclic. (In both theorems we assume that if $char K\ne p$ then $K$ contains a primitive $p^e$-th root of unity, where $p^e$ is the exponent of $G$.)

math.AG↗

arXiv2012

On realizability of $p$-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.

math.AG↗

arXiv2011

Noether's problem for central extensions of metacyclic $p$-groups

Let $K$ be a field and $G$ be a finite group. Let $G$ act on the rational function field $K(x(g):g\in G)$ by $K$ automorphisms defined by $g\cdot x(h)=x(gh)$ for any $g,h\in G$. Denote by $K(G)$ the fixed field $K(x(g):g\in G)^G$. Noether's problem then asks whether $K(G)$ is rational over $K$. In [M. Kang, Noether's problem for metacyclic $p$-groups, Adv. Math. 203(2005), 554-567], Kang proves the rationality of $K(G)$ over $K$ if $G$ is any metacyclic $p$-group and $K$ is any field containing enough roots of unity. In this paper, we give a positive answer to the Noether's problem for all central group extensions of the general metacyclic $p$-group, provided that $K$ is infinite and it contains sufficient roots of unity.

math.AG↗

arXiv2011

Noether's problem for some 2-groups

math.AG↗

arXiv2011

Noether's problem for the groups with a cyclic subgroup of index 4

Let $G$ be a finite group and $k$ be a field. Let $G$ act on the rational function field $k(x_g:g\in G)$ by $k$-automorphisms defined by $g\cdot x_h=x_{gh}$ for any $g,h\in G$. Noether's problem asks whether the fixed field $k(G)=k(x_g:g\in G)^G$ is rational (i.e. purely transcendental) over $k$. Theorem 1. If $G$ is a group of order $2^n$ ($n\ge 4$) and of exponent $2^e$ such that (i) $e\ge n-2$ and (ii) $ζ_{2^{e-1}} \in k$, then $k(G)$ is $k$-rational. Theorem 2. Let $G$ be a group of order $4n$ where $n$ is any positive integer (it is unnecessary to assume that $n$ is a power of 2). Assume that {\rm (i)} $\fn{char}k \ne 2$, $ζ_n \in k$, and {\rm (ii)} $G$ contains an element of order $n$. Then $k(G)$ is rational over $k$, except for the case $n=2m$ and $G \simeq C_m \rtimes C_8$ where $m$ is an odd integer and the center of $G$ is of even order (note that $C_m$ is normal in $C_m \rtimes C_8$) ; for the exceptional case, $k(G)$ is rational over $k$ if and only if at least one of $-1, 2, -2$ belongs to $(k^{\times})^2$.

math.AC↗