Related papers: Non-nilpotent subgroups of locally graded groups

Let $n$ be a positive integer and $\mathcal M$ a set of rational $n \times n$-matrices such that $\mathcal M$ generates a finite multiplicative semigroup. We show that any matrix in the semigroup is a product of matrices in $\mathcal M$…

Let $G$ be a finite soluble group and $h(G)$ its Fitting length. The aim of this paper is to give certain upper bounds for $h(G)$ as functions of the Fitting length of at least three Hall subgroups of $G$ which factorize $G$ in a particular…

A maximal abelian normal subgroup A in a nilpotent group N is self-centralizing. This makes their role an important one in determining the structure of the nilpotent group. For example if A is finite then N is also finite. In the free…

We consider ordered tuples in finite groups generating nilpotent subgroups. Given an integer $q$ we consider the poset of nilpotent subgroups of class less than $q$ and its corresponding coset poset. These posets give rise to a family of…

Let \(G\) be a finite group, and let \(\Delta(G)\) denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of \(G\). It is well known that, whenever \(\Delta(G)\) is connected, the diameter of…

We prove that $d(G) \log |G| = O(n^2 \log q)$ for irreducible subgroups $G$ of GL$(n,q)$, and estimate the associated constants. The result is motivated by attempts to bound the complexity of computing the automorphism groups of various…

Let $G$ be a finite group and $p$ a prime. We establish an upper bound for the derived length of a Sylow $p$-subgroup of $G$ in terms of the number of irreducible characters of $G$ whose degrees are divisible by $p$. We also prove that if…

We classify the finite groups whose non-linear irreducible characters that are not conjugate under the natural Galois action have distinct degrees, therefore extending the results in Berkovich et al. [Proc. Amer. Math. Soc. {\bf 115}…

The free nilpotent group $G_{m,n}$ of class $m$ and rank $n$ is the free object on $n$ generators in the category of nilpotent groups of class at most $m$. We show that $G_{m,n}$ can be recovered from its reduced group $C^*$-algebra, in the…

Let $\gamma_n=[x_1,\dots,x_n]$ be the $n$th lower central word. Denote by $X_n$ the set of $\gamma_n$-values in a group $G$ and suppose that there is a number $m$ such that $|g^{X_n}|\leq m$ for each $g\in G$. We prove that…

Finite non-abelian non-metacyclic $2$-generated $p$-groups (${p>2}$) of nilpotency class $2$ with cyclic commutator subgroup which are the additive groups of local nearrings are described. It is shown that the subgroup of all non-invertible…

In this paper we describe some properties of groups $G$ that contain a solvable subgroup of finite prime-power index (Theorem 1 and Corollaries 2--3). We prove that if $G$ is a non-solvable group that contains a solvable subgroup of index…

Let ${\Bbb Z}_{m}$ be the additive group of residue classes modulo $m$ and $s(m_{1},m_{2})$ denote the number of subgroups of the group ${\Bbb Z}_{m_{1}}\times {\Bbb Z}_{m_{2}}$, where $m_{1}$ and $m_{2}$ are arbitrary positive integers. We…

If $G$ and $H$ are finitely generated residually nilpotent groups, then $G$ and $H$ are in the same nilpotent genus if they have the same lower central quotients (up to isomorphism). A stronger condition is that $H$ is para-$G$ if there…

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…

The number of subgroups and the number of cyclic subgroups are natural combinatorial invariants of a finite group. We investigate how restrictions on these quantities, together with the number of distinct prime divisors of $|G|$, enforce…

This paper deals with the variety of commutative nonassociative algebras satisfying the identity $L_x^3+ \gamma L_{x^3} = 0$, $\gamma \in K$. Correa et al proved that if $\gamma = 0,1$ then any such finitely generated algebra is nilpotent.…

Let $G$ be a non-abelian $p$-group of order $p^n$ and $M(G)$ denote the Schur multiplier of $G$. Niroomand proved that $|M(G)| \leq p^{\frac{1}{2}(n+k-2)(n-k-1)+1}$ for non-abelian $p$-groups $G$ of order $p^n$ with derived subgroup of…

Let $G$ be a finite group and $p$ a fixed prime divisor of $|G|$. Combining the nilpotence, the normality and the order of groups together, we prove that if every maximal subgroup of $G$ is nilpotent or normal or has $p'$-order, then (1)…

Thompson's theorem stated that a finite group $G$ is solvable if and only if every $2$-generated subgroup of $G$ is solvable. In this paper, we prove some new criteria for both solvability and nilpotency of a finite group using certain…