Related papers: $L_9$-free groups

We classify the finite connected-homogeneous digraphs, as well as the infinite such digraphs with precisely one end. This completes the classification of all the locally finite connected-homogeneous digraphs.

In this paper we classify the finite groups satisfying the following property $P_4$: their orders of representatives are set-wise relatively prime for any 4 distinct non-central conjugacy classes.

We complete the classification of the finite special linear groups $\SL_n(q)$ which are $(2,3)$-generated, i.e., which are generated by an involution and an element of order $3$. This also gives the classification of the finite simple…

Let R be a finite unitary ring whose group of units is not solvable but all groups of units of all its proper subrings are solvable. In this paper we classify these rings and show that all finite rings of order $p^n$ for $n < 5$ and some of…

We investigate this class of groups originally called ulf (universal locally finite groups) of cardinality $\lambda$. We prove that for every locally finite group $G$ there is a canonical existentially closed extention of the same…

It is proven that an infinite finitely generated group cannot be elementarily equivalent to an ultraproduct of finite groups of a given Pr\"ufer rank. Furthermore, it is shown that an infinite finitely generated group of finite Pr\"ufer…

We prove that every {finitely generated residually finite}-by-sofic group satisfies Kaplansky's direct and stable finiteness conjectures with respect to all noetherian rings. We use this result to provide countably many new examples of…

We classify quasi-simple finite groups of essential dimension 3.

In this note we study a class of finite groups for which the orders of subgroups satisfy a certain inequality. In particular, characterizations of the well-known groups $\mathbb{Z}_2\times\mathbb{Z}_2$ and $S_3$ are obtained.

We introduce a class $\A$ of finitely generated residually finite accessible groups with some natural restriction on one-ended vertex groups in their JSJ-decompositions. We prove that the profinite completion of groups in $\A$ almost…

We characterize co-Hopfian finitely generated torsion free nilpotent groups in terms of their Lie algebra automorphisms, and construct many examples of such groups.

We study the class of groups having the property that every non-nilpotent subgroup is equal to its normalizer. These groups are either soluble or perfect. We completely describe the structure of soluble groups and finite perfect groups with…

We study groups having the property that every non-abelian subgroup is equal to its normalizer. This class of groups is closely related to an open problem posed by Berkovich. We give a full classification of finite groups having the above…

A finite presentation < X | R > of a finite group is called `just finite' if removing any relation from R results in a presentation for an infinite group. It has been an open question (Kourovka Notebook, Problem 21.10) whether every finite…

We prove that every finitely generated soluble group which is not virtually abelian has a subgroup of one of a small number of types.

We prove that groups for which every countable subgroup is free ($\aleph_1$-free groups) are n-slender, cm-slender, and lcH-slender. In particular every homomorphism from a completely metrizable group to an $\aleph_1$-free group has an open…

Given a finite group $G$, let $Cent(G)$ denote the set of distinct centralizers of elements of $G$. The group $G$ is called $n$-centralizer if $|Cent(G)|=n$ and primitive $n$-centralizer if $|Cent(G)|=|Cent(\frac{G}{Z(G)})|=n$. In this…

Let $G$ be a finite group, define $I(G)=\{x\in G : x^{2}=1\}$, $C(G)=$ set of the cyclic subgroups of $G$, $i(G)=|I(G)|$ and $c(G)=|C(G)|$. In this article, we will classify finite groups with $i(G)=c(G)-r$ for $r=0,1,$ and $2$. We also…

We give two examples of a finitely generated subgroup of a free group and a subset, closed in the profinite topology of a free group, such that their product is not closed in the profinite topology of a free group.

A finitely generated virtually free pro-p group with finite centralizers of its torsion elements is the free pro-p product of finite p-groups and a free pro-p factor.