Related papers: Solvable groups satisfying the two-prime hypothesi…

The spectrum $\omega(G)$ of a finite group $G$ is the set of orders of its elements. The following sufficient criterion of nonsolvability is proved: if among the prime divisors of the order of a group $G$, there are four different primes…

We study group-graded Lie algebras L with finite support X. We show that L is nilpotent of |X|-bounded class if X is arithmetically-free. Conversely: we show that Y supports the grading of a non-nilpotent Lie algebra if Y is not…

We find the nonabelian finite simple groups with order prime divisors not exceeding 1000. More generally, we determine the sets of nonabelian finite simple groups whose maximal order prime divisor is a fixed prime less than 1000. Our…

We will prove the following theorems. The first theorem posits the existence of a fixed point for the actions of nilpotent Lie groups on nonpositively curved compact manifolds. The second theorem states that actions of solvable Lie groups…

We describe the Reidemeister spectrum $Spec_RG$ for $G = N_{rc},$ the free nilpotent group of rank $r$ and class $c,$ in the cases: $r \in {\mathbb N}$ and $c = 1;$ $r = 2, 3$ and $c = 2;$ $ r = 2$ and $c = 3,$ and prove that any group…

Suppose that G is a finite group and x in G has prime order p > 3. Then x is contained in the solvable radical of G if (and only if) <x,x^g> is solvable for all g in G. If G is an almost simple group and x in G has prime order p > 3 then…

We prove the existence of abelian, solvable and nilpotent definable envelopes for groups definable in models of an NTP2 theory.

Let $G$ be a finite group and assume $p$ is a prime dividing the order of $G$. Suppose for any such $p$, that every two abelian $p$-subgroups of $G$ of equal order are conjugate. The structure of such a group $G$ has been settled in this…

We study a class of finite groups $G$ which behave similarly to elementary abelian $p$-groups with $p$ prime, that is, there exists a subgroup $N$ such that all elements of $G\setminus N$ are conjugate or inverse-conjugate under $\Aut(G)$.…

Let $G$ be a non-abelian group and $Z(G)$ be its center. The non-commuting graph $\mathcal{A}_G$ of $G$ is the graph whose vertex set is $G\backslash Z(G)$ and two vertices are joined by an edge if they do not commute. Let…

A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory…

Let $G$ be a finite group with the property that if $a,b$ are powers of $\delta_1^*$-commutators such that $(|a|,|b|)=1$, then $|ab|=|a||b|$. We show that $\gamma_{\infty}(G)$ is nilpotent.

The word $w=[x_{i_1},x_{i_2},\dots,x_{i_k}]$ is a simple commutator word if $k\geq 2, i_1\neq i_2$ and $i_j\in \{1,\dots,m\}$, for some $m>1$. For a finite group $G$, we prove that if $i_{1} \neq i_j$ for every $j\neq 1$, then the verbal…

Let p be a prime larger than 3 and let G be a finite group. We prove that G is p-solvable of p-length at most 2 if there are at most two distinct character degrees relatively prime to p in the principal p-block of G. This generalizes a…

Let G be a finite non-nilpotent group such that every Sylow subgroup of G is generated by at most d elements, and such that p is the largest prime dividing |G|. We show that G has a non-nilpotent image G/N, such that N is characteristic and…

In this note, I propose the following conjecture: a finite group G is nilpotent if and only if its largest quotient B-group \beta(G) is nilpotent. I give a proof of this conjecture under the additional assumption that G be solvable. I also…

Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of nonabelian simple groups. The minimum number of nonsolvable factors attained on all possible such series is called the…

This paper is an attempt to find out which properties of a finite group G can be expressed in terms of commutators of elements of coprime orders. A criterion of solubility of G in terms of such commutators is obtained. We also conjecture…

We introduce a combinatorial criterion for verifying whether a formula is not the conjunction of an equation and a co-equation. Using this, we give a proof for the nonequationality of the free group. Furthermore, we generalize the latter…

We show that any group $G$ is contained in some sharply 2-transitive group $\mathcal{G}$ without a non-trivial abelian normal subgroup. This answers a long-standing open question. The involutions in the groups $\mathcal{G}$ that we…