Related papers: Groups with the real chain condition on non-pronor…

For a group $G$ and a subgroup $H$ of $G$ this article discusses the normalizer of $H$ in the units of a group ring $RG$. We prove that $H$ is only normalized by the `obvious' units, namely products of elements of $G$ normalizing $H$ and…

The main theorem of Galois theory states that there are no finite group-subgroup pairs with the same invariants. On the other hand, if we consider complex linear reductive groups instead of finite groups, the analogous statement is no…

Any group that has a subnormal series, in which all factors are abelian and all except the last one are $p'$-torsion-free, can be embedded into a group with a subnormal series of the same length, with the same properties and such that any…

A group $G$ is said to have dense normalizers if each non-empty open interval in its subgroup lattice $L(G)$ contains the normalizer of a certain subgroup of $G$. In this note, we find all finite groups satisfying this property. We also…

Let $G$ be an affine algebraic group scheme over a field $k$. We show there exists a unipotent normal subgroup of $G$ which contains all other such subgroups; we call it the restricted unipotent radical $\mathrm{Rad}_u(G)$ of $G$. We…

Let G be a finite group. A subgroup M of G is said to be an NR-subgroup if, whenever K is normal in M, then K^G\cap M=K, where K^G is the normal closure of K in G. Using the Classification of Finite Simple Groups, we prove that if every…

If every element of a matrix group is similar to a permutation matrix, then it is called a permutation-like matrix group. References [4] and [5] showed that, if a permutation-like matrix group contains a maximal cycle of length equal to a…

A gyrogroup is a nonassociative group-like structure modelled on the space of relativistically admissible velocities with a binary operation given by Einstein's velocity addition law. In this article, we present a few of groups sitting…

Let $G$ be a finite non-cyclic group. The non-cyclic graph $\Gamma_G$ of $G$ is the graph whose vertex set is $G\setminus Cyc(G)$, two distinct vertices being adjacent if they do not generate a cyclic subgroup, where $Cyc(G)=\{a\in G:…

We prove that equivariantly simple invariant singularities can only exist for very few representations of a group of prime order: for real representations and some ``almost, but not quite real'' representations.

In the theory of flat chains with coefficients in a normed abelian group, we give a simple necessary and sufficient condition on a group element $g$ in order for the following fundamental regularity principle to hold: if a mass-minimizing…

We will determine all infinite $2$-locally finite groups as well as infinite $2$-groups with planar subgroup graph and show that infinite groups satisfying the chain conditions containing an involution do not have planar embeddings. Also,…

The order sequence of a finite group $G$ is a non-decreasing finite sequence formed of the element orders of $G$. Several properties of order sequences were studied by P. J. Cameron and H. K. Dey in a recent paper that concludes with a list…

In a finite group, a subset is called a Lagrange subset if its size divides the group order, and a factor if it admits a complementary subset. We provide a new and comparatively direct proof of the classification of groups in which every…

Suppose that $\Gamma$ is a non-empty connected graph, $\mathfrak{G}$ is the fundamental group of a graph of groups over $\Gamma$, and $\mathcal{C}$ is a root class of groups (the last means that $\mathcal{C}$ contains non-trivial groups and…

The residual closure of a subgroup $H$ of a group $G$ is the intersection of all virtually normal subgroups of $G$ containing $H$. We show that if $G$ is generated by finitely many cosets of $H$ and if $H$ is commensurated, then the…

Let $G$ be a finite group and $N$ a normal subgroup of $G$. We determine the structure of $N$ when the graph $\Gamma_G(N)$, which is the graph associated to the conjugacy classes of $G$ contained in $N$, has no triangles and when the graph…

Given a prime number \(p\) and a natural number \(m\) not divided by \(p\), we propose the problem of finding the smallest number \(r_{0}\) such that for \(r\geq r_{0}\), every group \(G\) of order \(p^{r}m\) has a non-trivial normal…

Let $t$ be a fixed natural number. A subgroup $H$ of a group $G$ will be called $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal in $G$ if there exists a chain of subgroups $H = H_{0} \leq H_{1} \leq \cdots \leq H_{m-1} \leq H_{m} = G$ such that…

A finite group G is called Schur, if every Schur ring over G is associated in a natural way with a regular subgroup of Sym(G) that is isomorphic to G. We prove that any nonabelian Schur group G is metabelian and the number of distinct prime…