Related papers: Non-nilpotent subgroups of locally graded groups

For a $p$-group of order $p^n$, it is known that the order of $2$-nilpotent multiplier is equal to $|\mathcal{M}^{(2)}(G)|=p^{\f12n(n-1)(n-2)+3-s_2(G)}$ for an integer $s_2(G)$. In this article, we characterize all of non abelian $p$-groups…

We consider groups of the nilpotency class $3$ of order $p^4$ which are the additive groups of local nearrings. It was shown that, for $p>3$, there exist a local nearring on one of such 4 groups.

Let $n>0$ be an integer and $\mathcal{X}$ be a class of groups. We say that a group $G$ satisfies the condition $(\mathcal{X},n)$ whenever in every subset with $n+1$ elements of $G$ there exist distinct elements $x,y$ such that $<x,y>$ is…

Let N be a normal subgroup of a finite group G and consider the set cd(G|N) of degrees of irreducible characters of G whose kernels do not contain N. A number of theorems are proved relating the set cd(G|N) to the structure of N. For…

The solvable Farb growth of a group quantifies how well-approximated the group is by its finite solvable quotients. In this note we present a new characterization of polycyclic groups which are virtually nilpotent. That is, we show that a…

We discuss multi-graded nilpotent tuples of multi-graded vector spaces which are a generalization of graded nilpotent pairs. The multi-grading yields a natural notion of a shape of such tuple and our main interest is to answer the question…

A recurring theme in finite group theory is understanding how the structure of a finite group is determined by the arithmetic properties of group invariants. There are results in the literature determining the structure of finite groups…

In this paper, we introduce the relative $n$-tensor nilpotent degree of a finite group $G$ with respect to a subgroup $H$ of $G$. The aim of this paper is to investigate this concept and give some results on this topic.

Let G be a finite solvable group, and let h(G) denote its Fitting height, namely the length of a shortest normal series in G with nilpotent factors. We show, that any law in G has length at least h(G). This result is then used to improve a…

Given a finite group $G$, we denote by $\nu(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup. We prove that if $\nu(G)>1/12,$ then $G$ is solvable.

We are interested in the algebraic properties of groups of local biholomorphisms and their consequences. A natural question is whether the complexity of solvable groups is bounded by the dimension of the ambient space. In this spirit we…

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…

The nonsoluble length $\lambda (G)$ of a finite group $G$ is defined as the number of nonsoluble factors in a shortest normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. The generalized…

For a finite group $G$, let $LC(G)$ be the subgroup generated by elements $x$ such that, for all $y \in G$ and all integers $n$, the order of $x^n y$ divides the least common multiple of the orders of $x$ and $y$. This subgroup is a…

We determine the structure of the finite non-solvable groups of order divisible by $3$ all whose maximal subgroups of order divisible by $3$ are supersolvable. Precisely, we demonstrate that if $G$ is a finite non-solvable group satisfying…

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…

Let M be a compact surface, either orientable or non-orientable. We study the lower central and derived series of the braid and pure braid groups of M in order to determine the values of n for which B\_n(M) and P\_n(M) are residually…

A semigroup is \emph{nilpotent} of degree 3 if it has a zero, every product of 3 elements equals the zero, and some product of 2 elements is non-zero. It is part of the folklore of semigroup theory that almost all finite semigroups are…

The nonsoluble length $\lambda(G)$ of a finite group $G$ is defined as the minimum number of nonsoluble factors in a normal series of $G$ each of whose quotients either is soluble or is a direct product of nonabelian simple groups. The…

Supersolubility of a finite group $G=\langle A,B\rangle$ with the nilpotent derived subgroup $G^\prime$ is established under the condition that the subgroups $A$ and $B$ are both subnormal and supersoluble.