Let $G$ be a finite group, $π$ be a set of primes, and define $H_π(G)$ to be the subgroup generated by all elements of $G$ which do not have prime order for every prime in $π$. In this paper, we investigate some basic properties of $H_π(G)$ and its relationship to the Hughes subgroup. We show that for most groups, only one of three possibilities occur: $H_π(G) = 1$, $H_π(G)=G$, or $H_π(G) = H_{p}(G)$ for some prime $p \in π$. There is only one other possibility: $G$ is a Frobenius group whose Frobenius complement has prime order $p$, and whose Frobenius kernel, $F$, is a nonabelian $q$-group such that $H_π(G)$ arises as a proper and nontrivial Hughes subgroup of $F$. We investigate a few restrictions on the possible choices of the primes $p$ and $q$.
Lévai and Pyber proposed the following as a conjecture: Let $G$ be a profinite group such that the set of solutions of the equation $x^n=1$ has positive Haar measure. Then $G$ has an open subgroup $H$ and an element $t$ such that all elements of the coset $tH$ have order dividing $n$ (see Problem 14.53 of [The Kourovka Notebook, No. 19, 2019]). \\ We define a constant $c_n$ for all finite groups and prove that the latter conjecture is equivalent with a conjecture saying $c_n<1$. Using the latter equivalence we observe that correctness of Lévai and Pyber conjecture implies the existence of the …
We prove that a uniform pro-p group with no nonabelian free subgroups has a normal series with torsion-free abelian factors. We discuss this in relation to unique product groups. We also consider generalizations of Hantzsche-Wendt groups.
Let $G$ be a group of order $n$ and $H$ be a subgroup of order $m$ of $G$. Denote by $ψ_H(G)$ the sum of element orders relative to $H$ of $G$. It is known that if $G$ is nilpotent, then $ψ_H(G)\leqψ_{H_m}(G)$, where $H_m$ is the unique subgroup of order $m$ of $C_n$. In this note, we show that this inequality does not hold for infinitely many finite solvable groups.
The study of verbal subgroups within a group is well-known for being an effective tool to obtain structural information about a group. Therefore, conditions that allow the classification of words in a free group are of paramount importance. One of the most studied problems is to establish which words are concise, where a word $w$ is said to be concise if the verbal subgroup $w(G)$ is finite in each group $G$ in which $w$ takes only a finite number of values. The purpose of this article is to present some results, in which a hierarchy among words is introduced, generalizing the concept of conci…
Yangming Li and Xianhua Li in 2012 proposed a conjecture that generalizes O.U. Kramer's result about supersoluble groups. Here we proved that this conjecture is false in the general case and true for groups with the trivial Frattini subgroup.
In this paper we study the groups all whose maximal or all Sylow subgroups are $K$-$\mathfrak{F}$-subnormal in their product the with generalizations of the Fitting subgroup $\mathrm{F}^*(G)$ and $\mathrm{\tilde F}(G)$. We prove that a hereditary formation $\mathfrak{F}$ contains every group all whose Sylow subgroups are $K$-$\mathfrak{F}$-subnormal in their product with $\mathrm{F}^*(G)$ if and only if $\mathfrak{F}$ is the class of all $σ$-nilpotent groups for some partition $σ$ of the set of all primes. We obtain a new characterization of the $σ$-nilpotent hypercenter, i.e. the $\mathfrak{F…
Given an Artin group $A_Γ$, a common strategy in the study of $A_Γ$ is the reduction to parabolic subgroups whose defining graphs have small diameter, i.e. showing that $A_Γ$ has a specific property if and only if all "small" parabolic subgroups of $A_Γ$ have this property. Since "small" parabolic subgroups are the puzzle pieces of $A_Γ$ one needs to study their behavior, in particular their intersections. The conjecture we address here says that the class of parabolic subgroups of $A_Γ$ is closed under intersection. Under the assumption that intersections of parabolic subgroups in complete Ar…
The integral Burau representation provides a map from the braid group into a group of integral matrices. This allows for a definition of congruence subgroups of the braid group as the preimage of the usual principal congruence subgroups of integral matrices. We explore the structure these congruence subgroups by examining some of the quotients that may arise in the series induced by divisibility of levels. We build on the work of Stylianakis on symmetric quotients of congruence subgroups, which itself generalizes the quotient of the braid group by the pure braid group. We accomplish this by ut…
Groups with a conjugacy class that is the difference of two normal subgroups
Kent State University（肯特州立大学）；University of York（约克大学）；Università degli Studi di Firenze（佛罗伦萨大学）；Universitat de València（瓦伦西亚大学）；Binghamton University（宾厄姆顿大学）
Let $ σ$ be a partition of the set of all primes and $\mathfrak{F}$ be a hereditary formation. We described all formations $\mathfrak{F}$ for which the $\mathfrak{F}$-hypercenter and the intersection of weak $K$-$\mathfrak{F}$-subnormalizers of all Sylow subgroups coincide in every group. In particular the formation of all $σ$-nilpotent groups has this property. With the help of our results we solve a particular case of L.A.~Shemetkov's problem about the intersection of $\mathfrak{F}$-maximal subgroups and the $\mathfrak{F}$-hypercenter. As corollaries we obtained P. Hall's and R. Baer's class…
In this article, we shall generalize a theorem due to Frobenius in group theory, which asserts that if $p$ is a prime and $p^{r}$ divides the order of a finite group, then the number of subgroups of order $p^{r}$ is $\equiv$ 1(mod $p$). Interestingly, our proof is purely combinatorial and does not use much group theory.
