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

Let p be a prime. Every finite group G has a normal series each of whose quotients either is p-soluble or is a direct product of nonabelian simple groups of orders divisible by p. The non-p-soluble length of G is defined as the minimal…

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this note, we prove that if $\sigma_1(G)<\frac{117}{20}$, then $G$ is solvable. Moreover, we have $\sigma_1(G)=\frac{117}{20}$ if and only if $G\cong A_5$.…

In this paper, we determine the structure of the nilpotent multipliers of all pairs $(G,N)$ of finitely generated abelian groups where $N$ admits a complement in $G$. Moreover, some inequalities for the nilpotent multipliers of pairs of…

For subsets $X,Y$ of a finite group $G$, let $Pr(X,Y)$ denote the probability that two random elements $x\in X$ and $y\in Y$ commute. Obviously, a finite group $G$ is nilpotent if and only if $Pr(P,Q)=1$ whenever $P$ and $Q$ are Sylow…

Following Isaacs (see [Isa08, p. 94]), we call a normal subgroup N of a finite group G large, if $C_G(N) \leq N$, so that N has bounded index in G. Our principal aim here is to establish some general results for systematically producing…

Let $FH$ be a supersolvable Frobenius group with kernel $F$ and complement $H$. Suppose that a finite group $G$ admits $FH$ as a group of automorphisms in such a manner that $C_G(F)=1$ and $C_{G}(H)$ is nilpotent of class $c$. We show that…

In this paper, we establish sharp thresholds on the independence number of the comaximal subgroup graph $\Gamma(G)$ that guarantee solvability, supersolvability, and nilpotency of the underlying group $G$. Specifically: \begin{itemize}…

The famous Tits' alternative states that a linear group either contains a nonabelian free group or is soluble-by-(locally finite). We study in this paper similar alternatives in pseudofinite groups. We show for instance that an…

In this paper, we study nilpotent $\mathbb{Q}$$[x]$-powered groups that satisfy the following property: For some set of primes $\omega$ in $\mathbb{Q}$$[x]$, every $\omega '$-isolated $\mathbb{Q}$$[x]$-subgroup in some family of its…

If G is a non-nilpotent group and nil(G) = {g \in G : <g, h> is nilpotent for all h\in G}, the nilpotent graph of G is the graph with set of vertices G-nil(G) in which two distinct vertices are related if they generate a nilpotent subgroup…

A group $G$ is said to be $n$-centralizer if its number of element centralizers $\mid \Cent(G)\mid=n$, an F-group if every non-central element centralizer contains no other element centralizer and a CA-group if all non-central element…

For a finite group $G$, let $d(G)$ denote the probability that a randomly chosen pair of elements of $G$ commute. We prove that if $d(G)>1/s$ for some integer $s>1$ and $G$ splits over an abelian normal nontrivial subgroup $N$, then $G$ has…

Let $G$ be a group and $Sol(G)=\{x \in G : \langle x,y \rangle \text{ is solvable for all } y \in G\}$. We associate a graph $\mathcal{NS}_G$ (called the non-solvable graph of $G$) with $G$ whose vertex set is $G \setminus Sol(G)$ and two…

The complexity of the equation solvability problem is known for nilpotent groups, for not solvable groups and for some semidirect products of Abelian groups. We provide a new polynomial time algorithm for deciding the equation solvability…

Let $G$ be a finite group and $N(G)$ be the set of conjugacy class sizes of $G$. For a prime $p$, let $|G||_p$ be the highest $p$-power dividing some element of $N(G)$. and define $|G|| = {\Pi}_{p\in {\pi}(G)}|G||_p$. $G$ is said to be an…

We show that there exists an algorithm to decide any single equation in the Heisenberg group in finite time. The method works for all two-step nilpotent groups with rank-one commutator, which includes the higher Heisenberg groups. We also…

Let $G$ be an odd order nilpotent group with class 2 and $e$ denotes the exponent of its commutator subgroup. Let $e=p_1^{r_1}p_2^{r_2}... p_s^{r_s}$, where $p_i$'s are odd primes and $r_i$'s are non-negative integers. Then there are at…

We refer to the set of the orders of elements of a finite group as its spectrum and say that groups are isospectral if their spectra coincide. We prove that with the only specific exception the solvable radical of a nonsolvable finite group…

We prove that any multiplicative subgroup G of the prime field f_p with |G| < p^{1/2} satisfies |3G| \gg |G|^2 / \log |G|. Also, we obtain a bound for the multiplicative energy of any nonzero shift of G, namely E^* (G+x) \ll |G|^2 log |G|,…

Let $\lambda(G)$ be the maximum number of subgroups in an irredundant covering of a finite group $G$. We prove that the finite groups with $\lambda(G)=|G|-t$, where $t\leq 5$, are solvable, and classify such groups.