Related papers: Subgroup separability in integral group rings

A finite group is said to be weakly separable if every algebraic isomorphism between two $S$-rings over this group is induced by a combinatorial isomorphism. In the paper we prove that every abelian weakly separable group belongs to one of…

We classify finite groups $G$, such that the group algebra, $\mathbb{Q}G$ (over the field of rational numbers $\mathbb{Q}$), is the direct product of the group algebra $\mathbb{Q}[G/N]$ of a proper factor group $G/N$, and some division…

Let $p$ be a prime number and suppose that every maximal subgroup of a finite group is either $p$-nilpotent or has prime index. Such group need not be $p$-solvable, and we study its structure by proving that only one nonabelian simple group…

Let $G$ be a sp-group such that for every prime $p$, $G_p$ is elementary. %$\oplus \End_{\zz}(G_p) \leq \End_{\zz}(G) \leq \prod \End_{\zz}(G_p)$. Suppose that $\frac{G}{\oplus_{p\in \mathbb{P}} G_p}$ is torsion-free divisible. %In this…

An abstract group $G$ is called totally $2$-closed if $H=H^{(2),\Omega}$ for any set $\Omega$ with $G\cong H\leq{\rm Sym}(\Omega)$, where $H^{(2),\Omega}$ is the largest subgroup of ${\rm Sym}(\Omega)$ whose orbits on $\Omega\times\Omega$…

Elementarily free groups are the finitely generated groups with the same elementary theory as free groups. We prove that elementarily free groups are subgroup separable, answering a question of Zlil Sela.

Let $G$ denote the projective special linear group $\text{PSL}(2,q)$, for a prime power $q$. It is shown that a finite 2-subgroup of the group $V(\mathbb{Z}G)$ of augmentation 1 units in the integral group ring $\mathbb{Z}G$ of $G$ is…

Suppose $C(G)$ denotes the set of all cyclic subgroups of a finite group $G$, and $\mathcal{O}_{2}(G)$ denotes the number of elements of order $2$ in $G$. In [Marius T., Finite groups with a certain number of cyclic subgroups. The American…

We show that $\mathcal{U}(\mathbb{Z}G)$, the unit group of the integral group ring $\mathbb{Z} G$, either satisfies Kazhdan's property (T) or is, up to commensurability, a non-trivial amalgamated product, in case $G$ is a finite group…

We give a new, effective proof of the separability of cubically convex-cocompact subgroups of special groups. As a consequence, we show that if $G$ is a virtually compact special hyperbolic group, and $Q\leq G$ is a $K$-quasiconvex…

A group $G$ is called subgroup conjugacy separable (abbreviated as SCS), if any two finitely generated and non-conjugate subgroups of $G$ remain non-conjugate in some finite quotient of $G$. We prove that free groups and the fundamental…

Suppose that $G$ is a finite solvable group and $V$ is a finite, faithful and completely reducible $G$-module. Let $N$ be a nilpotent subgroup of $G$, then there exits $v \in V$ such that $|\bC_N(v)| \leq (|N|/p)^{1/p}$, where $p$ is the…

Let $D$ be a division ring with infinite center, $K$ a proper division subring of $D$ and $N$ an almost subnormal subgroup of the multiplicative group $D^*$ of $D$. The aim of this paper is to show that if $K$ is $N$-invariant and $N$ is…

Let G be an isotropic reductive algebraic group over a commutative ring R. Assume that the elementary subgroup E(R) of group of points G(R) is correctly defined. Then E(R) is perfect, except for the well-known cases of a split reductive…

We prove that if a group $G$ admits a virtually special action on a CAT(0) cube complex, then any product of convex-cocompact subgroups of $G$ is separable. Previously, this was only known for products of three subgroups, or in the case…

We classify all finite groups G such that the product of any two non-inverse conjugacy classes of G is always a conjugacy class of G. We also classify all finite groups G for which the product of any two G-conjugacy classes which are not…

Let the group $G = AB$ be the product of subgroups $A$ and $B$, and let $p$ be a prime. We prove that $p$ does not divide the conjugacy class size (index) of each $p$-regular element of prime power order $x\in A\cup B$ if and only if $G$ is…

In this paper, it is proved that the group generated by Bass units contains a subgroup of finite index in the group of central units $\mathcal{Z}(\mathcal{U}(\mathbb{Z}G))$ of the integral group ring $\mathbb{Z}G$ for a subgroup closed…

For a finite group $G$ we investigate the difference between the maximum size MaxDim$(G)$ of an "independent" family of maximal subgroups of $G$ and maximum size $m(G)$ of an irredundant sequence of generators of $G$. We prove that…

Let $M$ be a maximal subgroup of a finite group $G$ and $K/L$ be a chief factor such that $L\leq M$ while $K\nsubseteq M$. We call the group $M\cap K/L$ a $c$\ns section of $M$. And we define $Sec(M)$ to be the abstract group that is…