arXiv subjects
Publications and source records attributed to Marius Tărnăuceanu.
In this paper, we describe the structure of finite groups whose element orders or proper (abelian) subgroup orders form an arithmetic progression of ratio $r\geq 2$. This extends the case $r=1$ studied in previous papers \cite{1,8,4}.
Let $G$ be a finite group and $α(G)=\frac{|C(G)|}{|G|}$\,, where $C(G)$ denotes the set of cyclic subgroups of $G$. In this short note, we prove that $α(G)\leqα(Z(G))$ and we describe the groups $G$ for which the equality occurs. This gives some sufficient conditions for a finite group to be $4$-abelian or abelian.
Let $C(G)$ be the poset of cyclic subgroups of a finite group $G$ and let $\mathcal{P}$ be the class of $p$-groups of order $p^n$ ($n\geq 3$). Consider the function $α:\mathcal{P}\longrightarrow (0, 1]$ given by $α(G)=\frac{|C(G)|}{|G|}$. In this paper, we determine the second minimum value of $α$, as well as the corresponding minimum points. Further, since the problem of finding the second maximum value of $α$ was completely solved for $p=2$, we focus on the case of odd primes and we outline a result in this regard.
Let $G$ be a finite group and let $ψ(G)$ denote the sum of element orders of $G$. It is well-known that the maximum value of $φ$ on the set of groups of order $n$, where $n$ is a positive integer, will occur at the cyclic group $C_n$. For nilpotent groups, we prove a natural generalization of this result, obtained by replacing the element orders of $G$ with the element orders relative to a certain subgroup $H$ of $G$.
Let $k$ be a positive integer and $G$ be a finite group that cannot be written as the union of $k$ proper subgroups. In this short note, we study the existence of a constant $c_k\in (0,1)$ such that $|\cup_{i=1}^k H_i| \leq c_k|G|$, for all proper subgroups $H_1$, ..., $H_k$ of $G$.
Let $G$ be a finite group and $σ_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this note, we prove that if $σ_1(G)<\frac{117}{20}$, then $G$ is solvable. Moreover, we have $σ_1(G)=\frac{117}{20}$ if and only if $G\cong A_5$. This solves an open problem posed in \cite{8}.
In this paper, we introduce a new function related to the sum of element orders of finite groups. It is used to give some criteria for a finite group to be cyclic, abelian, nilpotent, supersolvable and solvable, respectively.
Denote the sum of element orders in a finite group $G$ by $ψ(G)$ and let $C_n$ denote the cyclic group of order $n$. In this paper, we prove that if $|G|=n$ and $ψ(G)>\frac{13}{21}\,ψ(C_n)$, then $G$ is nilpotent. Moreover, we have $ψ(G)=\frac{13}{21}\,ψ(C_n)$ if and only if $n=6m$ with $(6,m)=1$ and $G\cong S_3\times C_m$. Two interesting consequences of this result are also presented.
In this note, we study the finite groups whose Chermak-Delgado measure has exactly two values. They determine an interesting class of $p$-groups containing cyclic groups of prime order and extraspecial $p$-groups.
In this note, we prove that $D_8\times C_2^{n-3}$ is the non-elementary abelian $2$-group of order $2^n$, $n\geq 3$, whose number of subgroups of possible orders is maximal. This solves a conjecture by Haipeng Qu [7]. A formula for counting the subgroups of an (almost) extraspecial $2$-group is also presented.
The aim of the present paper is to study isomorphisms of canonical ${\cal E}$-lattices. Some interesting results are obtained in the particular case of isomorphisms between two subgroup ${\cal E}$-lattices.
In the present paper we introduce and study a canonical ${\cal E}$-lattice structure on the set of element orders of some finite groups. We show that a finite abelian group is uniquely determined by this canonical ${\cal E}$-lattice.
In this note, we give a new formula for the number of cyclic subgroups of a finite abelian group. This is based on applying the Burnside's lemma to a certain group action. Also, it generalizes the well-known Menon's identity.
In this note we introduce and characterize a class of finite groups for which the element orders satisfy a certain inequality. This is contained in some well-known classes of finite groups.
The main goal of this note is to provide a new proof of a classical result about projectivities between finite abelian groups. It is based on the concept of fundamental group lattice, studied in our previous papers \cite{8} and \cite{9}. A generalization of this result is also given.
Given a finite group $G$, we denote by $ψ\,'(G)$ the product of element orders of $G$. Our main result proves that the restriction of $ψ\,'$ to abelian $p$-groups of order $p^n$ is strictly increasing with respect to a natural order on the groups relating to the lexicographic order of the partitions of $n$. In particular, we infer that two finite abelian groups of the same order are isomorphic if and only if they have the same product of element orders.
In this note we give a bijective proof for the explicit formula giving the total number of principal series of the direct product $\mathbb{Z}_{p^{α_1}} \times \mathbb{Z}_{p^{α_2}}$, where $p$ is a prime number. This new proof is easier to generalize to arbitrary finite abelian groups than the original direct calculation method.
In this article we introduce and study a class of finite groups for which the orders of normal subgroups satisfy a certain inequality. It is closely connected to some well-known arithmetic classes of natural numbers.