Related papers: Locally finite groups in which every non-cyclic su…

A group $G$ is said to have dense solitary subgroups if each non-empty open interval in its subgroup lattice $L(G)$ contains a solitary subgroup. In this short note, we find all finite groups satisfying this property.

We introduce and investigate a class of profinite groups defined via extensions of centralizers analogous to the extensively studied class of finitely generated fully residually free groups, that is, limit groups (in the sense of Z. Sela).…

A finite group $G$ is called $k$-factorizable if for every ordered factorization $|G|=a_1\cdots a_k$ into integers each greater than $1$ there exist subsets $A_1,\dots,A_k\subseteq G$ such that $|A_i|=a_i$ for each $i$ and $G=A_1\cdots…

Every countable group $G$ can be embedded in a finitely generated group $G^*$ that is hopfian and complete, i.e. $G^*$ has trivial centre and every epimorphism $G^*\to G^*$ is an inner automorphism. Every finite subgroup of $G^*$ is…

A group is said to be stable if it is isomorphic to its automorphism group. We investigate how we can extend centerless groups to construct finite stable groups with nontrivial centers. To this end, we classify all finite stable groups…

In this note we introduce and characterize a class of finite groups for which the element orders satisfy a certain inequality. This is contained in some well-known classes of finite groups.

A finite group is called a CLT-group if it contains a subgroup corresponding to every divisor of the order of the group. It is said to be a Cyclic (Abelian) CLT group if it contains a cyclic (abelian) subgroup corresponding to every proper…

An automorphism $\alpha$ of a group $G$ is said to be central if $\alpha$ commutes with every inner automorphism of $G$. We construct a family of non-special finite $p$-groups having abelian automorphism groups. These groups provide counter…

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms. For any fixed prime divisor $p$ of $|G|$, we provide a complete characterization of the structure of a group $G$ in which every maximal $A$-invariant…

We discuss the finiteness of the topological entropy of continuous endomorphims for some classes of locally compact groups. Firstly, we focus on the abelian case, imposing the condition of being compactly generated, and note an interesting…

Given a prime power $p^d$ with $p$ a prime and $d$ a positive integer, we classify the finite groups $G$ with $p^{2d}$ dividing $|G|$ in which all subgroups of order $p^d$ are complemented and the finite groups $G$ having a normal…

This is one of a series of papers which aims towards a classification of imprimitive affine groups of rank $3$. In this paper, a complete classification is given of such groups of characteristic $p$ such that the point stabilizer is not…

In this note, we show that an uncountable locally free group, and therefore every locally free group, has a free subgroup whose cardinality is the same as that of $G$. This result directly improve the main result in [T. Nishinaka,"Group…

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,…

We continue in this paper the study of locally minimal groups started in \cite{LocMin}. The minimality criterion for dense subgroups of compact groups is extended to local minimality. Using this criterion we characterize the compact abelian…

We study finite groups which possess a strongly p-embedded subgroup for some odd prime p. The main results of the paper will be applied in the ongoing project to classify the simple groups of local characteristic p.

We prove that a permutation group in which different finite sets have different stabilizers cannot satisfy any group law. For locally compact topological groups with this property we show that almost all finite subsets of the group generate…

The set of all closed subgroups of a profinite carries a natural profinite topology. This space of subgroups can be classified up to homeomorphism in many cases, and tight bounds placed on its complexity as expressed by its scattered…

Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. Let $\mathrm{Aut}_c(G)$ and $\mathrm{Aut}_z(G)$ respectively denote the group of all class preserving and central automorphisms of $G$. We give a necessary condition for $G$…

Let $G$ be a finite non-abelian group and $m=|G|/|Z(G)|$. In this paper we investigate $m$-centralizer group $G$ with cyclic center and we will prove that if $G$ is a finite non-abelian $m$-centralizer $CA$-group, then there exists an…