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Mihai-Silviu Lazorec

Publications and source records attributed to Mihai-Silviu Lazorec.

At least 19 recordsLinked to original sources
arXiv2026

On the number of conjugacy classes of subgroups of a finite group

Let $k'(G)$ and $L(G)$ be the number of conjugacy classes of subgroups and the subgroup lattice of a finite group $G$, respectively. Our objective is to study some aspects related to the ratios $d'(G)=\frac{k'(G)}{|L(G)|}$ and $d^*(G)=\min\{ d'(S) \mid S\text{ is a section of }G\}$ which measure how close $G$ is to being a Dedekind group. We prove that the set containing the values $d'(G)$, as $G$ ranges over the class of nilpotent groups, is dense in $[0, 1]$. A nilpotency criterion is obtained by proving that if $d^*(G)>\frac{2}{3}$, then $G$ is nilpotent and information on its structure is given. We also show that if $d^*(G)>\frac{4}{5}$, then $G$ is an Iwasawa group. Finally, we deduce a result which ensures that a $p$-group of order $p^n$ ($n\geq 3$) is a Dedekind group. This last result can be extended to the class of nilpotent groups and it also highlights the second maximum value of $d^*$ on the class of $p$-groups of order $p^n$.

math.GR↗
arXiv2026

Detecting solvability, supersolvability and CLT properties via an invariant based on conjugacy classes of subgroups

For a finite group $G$, denote by $k'(G)$ and $L(G)$ the number of conjugacy classes of subgroups and the subgroup lattice of $G$, respectively. Let $d'(G)=\frac{k'(G)}{|L(G)|}$ and $d^*(G)$ be the minimum value of $d'(S)$, when $S$ runs through all sections of $G$. In this paper we deduce some criteria on the nature of $G$. We show that if $d^*(G)>\frac{9}{59}$, then $G$ is solvable, while if $d^*(G)>\frac{1}{2}$, then $G$ is a supersolvable group. The last criterion is also valid when replacing ``supersolvable" with ``CLT".

math.GR↗
arXiv2026

A set theoretic version of equations on groups

Let $G$ be a finite group. The aim of this paper is to study the number of solutions $S\subseteq G$ of the equation $\mho^{\{n\}}(S)=L$, where $L$ is a non-empty subset of $G$, $n$ is a positive integer and $\mho^{\{n\}}(S)=\{ s^n \ | \ s\in S\}$. Besides our findings obtained in this general frame, we also outline some results which hold for some particular cases such as: \textit{i)} $L$ is a normal subset of $G$; \textit{ii)} $G$ is abelian; \textit{iii)} $G$ is an extraspecial $p$-group.

math.GR↗
arXiv2026

Minimum $\mathcal{F}$-covers: the supersolvable and metabelian cases

Given a set $\mathcal{F}$ of finite groups, it is said that a group $G$ is an $\mathcal{F}$-cover if every group in $\mathcal{F}$ is isomorphic to a subgroup of $G$. Moreover, $G$ is a minimum $\mathcal{F}$-cover if there is no $\mathcal{F}$-cover whose order is less than $|G|$. In [Cameron P. J., et al., Minimal cover groups, J. Algebra 660 (2024)], the authors pose the following question: For which classes $\mathcal{X}$ of groups, closed under taking subgroups and direct products, is it true that, if $\mathcal{F}$ is a set of $\mathcal{X}$-groups, then there is a minimum $\mathcal{F}$-cover which is an $\mathcal{X}$-group? In this paper, we give a negative answer in two cases: $\mathcal{X}\in \{``supersolvable", ``metabelian"\}.$

math.GR↗
arXiv2026

On some problems regarding $LCM$-groups

Let $G$ be a finite group and denote by $o(g)$ the order of an element $g\in G$. We say that $G$ is an $LCM$-group if $o(x^ny)$ is a divisor of the least common multiple of $o(x^n)$ and $o(y)$ for all $x, y\in G$ and $n\in\mathbb{N}$. This paper extends some existing results on $LCM$-groups, such as the structure of a minimal non-$LCM$-group, and establishes criteria for $G$ to be an $LCM$-group or a nilpotent group. We also prove that, in general, a minimum cover of a finite set of $LCM$-groups is not an $LCM$-group, and we answer two questions posed by M. Amiri.

math.GR↗
arXiv2024

On two problems about order sequences of finite groups

The order sequence of a finite group $G$ is a non-decreasing finite sequence formed of the element orders of $G$. Several properties of order sequences were studied by P. J. Cameron and H. K. Dey in a recent paper that concludes with a list of open problems. In this paper we solve two of these problems by showing the following facts: 1) if there is a non-supersolvable/non-solvable group of order $n$, it is not always true that its order sequence is properly dominated by the order sequence of any supersolvable/solvable group of order $n$; 2) the supersolvability of a finite group cannot be described by its order sequence.

math.GR↗
arXiv2024

Element orders in extraspecial groups

By using the structure and some properties of extraspecial and generalized/almost extraspecial $p$-groups, we explicitly determine the number of elements of specific orders in such groups. As a consequence, one may find the number of cyclic subgroups of any (generalized/almost) extraspecial group. For a finite group $G$, the ratio of the number of cyclic subgroups to the number of subgroups is called the cyclicity degree of $G$ and is denoted by $cdeg(G)$. We show that the set containing the cyclicity degrees of all finite groups is dense in $[0, 1]$. This is equivalent to giving an affirmative answer to the following question posed by Tóth and Tărnăuceanu: ``For every $a\in [0, 1]$, does there exist a sequence $(G_n)_{n\geq 1}$ of finite groups such that $\displaystyle\lim_{n\to\infty} cdeg(G_n)=a$?". We show that such sequences are formed of finite direct products of extraspecial groups of a specific type.

math.GR↗
arXiv2024

Subgroups with a small sum of element orders

A finite group $G$ is said to be a $\mathcal{B}_ψ$-group if $ψ(H)<|G|$ for any proper subgroup $H$ of $G$, where $ψ(H)$ denotes the sum of element orders of $H$. In this paper, we characterize the $\mathcal{B}_ψ$-groups up to the class of finite nilpotent groups. We also study the above inequality in the case of finite $P$-groups and Schmidt groups, respectively.

math.GR↗
arXiv2023

Some density results involving the average order of a finite group

Let $o(G)$ be the average of the element orders of a finite group $G$. A research topic concerning this quantity is understanding the relation between $o(G)$ and $o(H)$, where $H$ is a subgroup of $G$. Let $\mathcal{N}$ be the class of finite nilpotent groups and let $L(G)$ be the subgroup lattice of $G$. In this paper, we show that the set $\lbrace \frac{o(G)}{o(H)} \ | \ G\in\mathcal{N}, H\in L(G)\rbrace$ is dense in $[0, \infty)$. Other density results are outlined throughout the paper.

math.GR↗
arXiv2022

On the average order of a finite group

Let $o(G)$ be the average order of a finite group $G$. We show that if $o(G)<c$, where $c\in \lbrace \frac{13}{6}, \frac{11}{4}\rbrace$, then $G$ is an elementary abelian 2-group or a solvable group, respectively. Also, we prove that the set containing the average orders of all finite groups is not dense in $[a, \infty)$, for all $a\in [0, \frac{13}{6}]$. We also outline some results related to the integer values of the average order. Since group element orders is a popular research topic, we pose some open problems concerning the average order of a finite group throughout the paper.

math.GR↗
arXiv2022

A graph related to the sum of element orders of a finite group

A finite group is called $ψ$-divisible iff $ψ(H)|ψ(G)$ for any subgroup $H$ of a finite group $G$. Here, $ψ(G)$ is the sum of element orders of $G$. For now, the only known examples of such groups are the cyclic ones of square-free order. The existence of non-abelian $ψ$-divisible groups still constitutes an open question. The aim of this paper is to make a connection between the $ψ$-divisibility property and graph theory. Hence, for a finite group $G$, we introduce a simple undirected graph called the $ψ$-divisibility graph of $G$. We denote it by $ψ_G$. Its vertices are the non-trivial subgroups of $G$, while two distinct vertices $H$ and $K$ are adjacent iff $H\subset K$ and $ψ(H)|ψ(K)$ or $K\subset H$ and $ψ(K)|ψ(H)$. We prove that $G$ is $ψ$-divisible iff $ψ_G$ has a universal (dominating) vertex. Also, we study various properties of $ψ_G$, when $G$ is a finite cyclic group. The choice of restricting our study to this specific class of groups is motivated in the paper.

math.GR↗
arXiv2020

Groups whose same-order types are arithmetic progressions

The same-order type $τ_e(G)$ of a finite group $G$ is a set formed of the sizes of the equivalence classes containing the same order elements of $G$. In this paper, we study an arithmetical property of this set. More exactly, we outline some results on the classification and existence of finite groups whose same-order types are arithmetic progressions formed of 3 or 4 elements, the latter being the maximum size of such a sequence.

math.GR↗
arXiv2020

On a divisibility property involving the sum of element orders

A finite group $G$ is called $ψ$-divisible if $ψ(H)|ψ(G)$ for any subgroup $H$ of $G$, where $ψ(H)$ and $ψ(G)$ are the sum of element orders of $H$ and $G$, respectively. In this paper, we extend a result provided in [10], by classifying the finite groups whose all subgroups are $ψ$-divisible. Since the existence of $ψ$-divisible groups is related to the class of square-free order groups, we also study the sum of element orders and the $ψ$-divisibility property of ZM-groups. In the end, we introduce the concept of $ψ$-normal divisible group, i.e. a group for which the $ψ$-divisibility property is satisfied by all its normal subgroups. Using simple and quasisimple groups, we are able to construct infinitely many $ψ$-normal divisible groups which are neither simple nor nilpotent.

math.GR↗
arXiv2020

The second minimum/maximum value of the number of cyclic subgroups of finite $p$-groups

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.

math.GR↗
arXiv2019

A density result on the sum of element orders of a finite group

Let $\mathcal{G}$ be the class of all finite groups and consider the function $ψ'':\mathcal{G}\longrightarrow(0,1]$, given by $ψ''(G)=\frac{ψ(G)}{|G|^2}$, where $ψ(G)$ is the sum of element orders of a finite group $G$. In this paper, we show that the image of $ψ''$ is a dense set in $[0, 1]$. Also, we study the injectivity and the surjectivity of $ψ''$.

math.GR↗
arXiv2019

A connection between the number of subgroups and the order of a finite group

For a finite group $G$, we associate the quantity $β(G)=\frac{|L(G)|}{|G|}$, where $L(G)$ is the subgroup lattice of $G$. Different properties and problems related to this ratio are studied throughout the paper. We determine the second minimum value of $β$ on the class of $p$-groups of order $p^n$, where $n\geq 3$ is an integer. We show that the set containing the quantities $β(G)$, where $G$ is a finite (abelian) group, is dense in $[0,\infty).$ Finally, we consider $β$ to be a function on $L(G)$ and we mark some of its properties, the main result being the classification of finite abelian $p$-groups $G$ satisfying $β(H)\leq 1, \ \forall \ H\in L(G).$

math.GR↗
arXiv2018

A note on the number of cyclic subgroups of a finite group

Let $G$ be a finite group, $L_1(G)$ be its poset of cyclic subgroups and consider the quantity $α(G)=\frac{|L_1(G)|}{|G|}$. The aim of this paper is to study the class $\cal{C}$ of finite nilpotent groups having $α(G)=\frac{3}{4}$. We show that if $G$ belongs to this class, then it is a 2-group satisfying certain conditions. Also, we study the appartenance of some classes of finite groups to $\cal{C}$.

math.GR↗