Related papers: On finite $d$-maximal groups

For a finite group $G$ and an element $x\in G$, the subset $$ nil_G(x)=\{y\in G \mid <x,y>~~ is ~~ nilpotent\}$$ is called nilpotentizer of $x$ in $G$. In this paper, we give two solvabilty criteria for a finite group by the structure and…

The isomorphism problem for infinite finitely presented groups is probably the hardest among standard algorithmic problems in group theory. Classes of groups where it has been completely solved are nilpotent groups, hyperbolic groups, and…

A subset S of a group G invariably generates G if G = <s^(g(s)) | s in S> for each choice of g(s) in G, s in S. In this paper we study invariable generation of infinite groups, with emphasis on linear groups. Our main result shows that a…

A group is called capable if it is a central factor group. For each prime $p$ and positive integer $c$, we prove the existence of a capable $p$-group of class $c$ minimally generated by an element of order $p$ and an element of order…

We prove:(1) the existence, for every integer n > 3, of a noncompact smooth n-dimensional topological manifold whose diffeomorphism group contains an isomorphic copy of every finitely presented group; (2) a finiteness theorem on finite…

T.C. Burness and S.D. Scott \cite{3} classified finite groups $G$ such that the number of prime order subgroups of $G$ is greater than $|G|/2-1$. In this note, we study finite groups $G$ whose subgroup graph contains a vertex of degree…

We describe the class of graphs for which all metric spaces with diametrical graphs belonging to this class are ultrametric. It is shown that a metric space $(X, d)$ is ultrametric iff the diametrical graph of the metric $d_{\varepsilon}(x,…

We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite…

Let $w = w(x_1,..., x_n)$ be a word, i.e. an element of the free group $F =<x_1,...,x_n>$ on $n$ generators $x_1,..., x_n$. The verbal subgroup $w(G)$ of a group $G$ is the subgroup generated by the set $\{w (g_1,...,g_n)^{\pm 1} | g_i \in…

Let $G$ be a finite group. If $M_n < M_{n-1} < \ldots < M_1 < M_{0}=G $ where $M_i$ is a maximal subgroup of $M_{i-1}$ for all $i=1, \ldots ,n$, then $M_n $ ($n > 0$) is an \emph{$n$-maximal subgroup} of $G$. A subgroup $M$ of $G$ is called…

We investigate the two classes of finite subgroups of SU(3) that are called type C and D in the book of Miller, Blichfeldt and Dickson. We present two theorems which fully determine the form of the generators in a suitable basis. After…

Let $G$ be a finite group and $k$ an algebraically closed field of characteristic $p>0$. We define the notion of a Dade $kG$-module as a generalization of endo-permutation modules for $p$-groups. We show that under a suitable equivalence…

In this paper, we assume that $G$ is a finitely generated torsion free non-elementary Kleinian group with $\Omega(G)$ nonempty. We show that the maximal number of elements of $G$ that can be pinched is precisely the maximal number of rank 1…

Many common finite p-groups admit automorphisms of order coprime to p, and when p is odd, it is reasonably difficult to find finite p-groups whose automorphism group is a p-group. Yet the goal of this paper is to prove that the automorphism…

If a finite group G has a presentation with d generators and r relations, it is well-known that r - d is at least the rank of the Schur multiplier of G; a presentation is called efficient if equality holds. There is an analogous definition…

A group is properly 3-realizable if it is the fundamental group of a compact polyhedron whose universal covering is proper homotopically equivalent to some 3-manifold. We prove that when such a group is also quasi-simply filtered then it…

We continue our earlier study of finite dimensional definable groups in models of the the model companion of an o-minimal L-theory T expanded by a generic derivation as in [F-K]. We generalize Buium's notion of an algebraic D-group to…

In this paper we investigate the following general problem. Let $G$ be a group and let $i(G)$ be a property of $G$. Is there an integer $d$ such that $G$ contains a $d$-generated subgroup $H$ with $i(H)=i(G)$? Here we consider the case…

A finite group R is a CI-group if, whenever S and T are subsets of R with the Cayley graphs Cay(R,S) and Cay(R,T) isomorphic, there exists an automorphism x of R with S^x=T. The classification of CI-groups is an open problem in the theory…

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…