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Marius Tărnăuceanu

Publications and source records attributed to Marius Tărnăuceanu.

At least 19 recordsLinked to original sources
arXiv2026

Two new functions related to the sum of element orders of a finite group

Let $G$ be a finite group and $H$ be a proper normal subgroup of $G$. In this note, we introduce two new functions $o(G,H)$ and $ψ''(G,H)$ involving the sums of element orders of $G$ and $H$. We prove that they satisfy certain inequalities, with equality if and only if $(G,H)$ is an equal order pair of a particular type.

math.GR↗
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↗
arXiv2025

The Reidemeister spectrum of ZM-groups

Given a group $G$ and an automorphism $φ$ of $G$, two elements $x,y\in G$ are said to be $φ$-conjugate if $x=gyφ(g)^{-1}$ for some $g\in G$. The number $R(φ)$ of equivalence classes with respect to this relation is called the Reidemeister number of $φ$ and the set $\{R(φ)|φ\in {\rm Aut}(G)\}$ is called the Reidemeister spectrum of $G$. In this paper, we determine the Reidemeister spectrum of ZM-groups, extending some results of \cite{13}.

math.GR↗
arXiv2025

Almost and quasi Leinster groups

In this paper, we study the parallelism between perfect numbers and Leinster groups and continue it by introducing the new concepts of almost and quasi Leinster groups which parallel almost and quasi perfect numbers. These are small deviations from perfect numbers; very few results and/or examples are known about them. We investigate nilpotent almost-/quasi-/Leinster groups and find some examples and conditions for the existence of such groups for classes of non-nilpotent groups: ZM (Zassenhaus metacyclic) groups, dihedral generalised groups, generalised dyciclic groups and affine groups.

math.GR↗
arXiv2025

A result on certain sums of element orders in finite groups

Given a finite group $G$ of order $p^nm$, where $p$ is a prime and $p\nmid m$, we denote by $ψ_p(G)$ the sum of orders of $p$-parts of elements in $G$. In the current note, we prove that $ψ_p(G)\leqψ_p(C_{p^nm})$, where $C_{p^nm}$ is the cyclic group of order $p^nm$, and the equality holds if and only if $G$ is $p$-nilpotent of a particular type. A generalization of this result is also presented.

math.GR↗
arXiv2025

Finite groups with many normalizers

A group $G$ is said to have dense normalizers if each non-empty open interval in its subgroup lattice $L(G)$ contains the normalizer of a certain subgroup of $G$. In this note, we find all finite groups satisfying this property. We also classify the finite groups in which $k$ subgroups are not normalizers, for $k=1,2,3,4$.

math.GR↗
arXiv2025

On the Chermak-Delgado lattice of a finite group

By imposing conditions upon the index of a self-centralizing subgroup of a group, and upon the index of the center of the group, we are able to classify the Chermak-Delgado lattice of the group. This is our main result. We use this result to classify the Chermak-Delgado lattices of dicyclic groups and of metabelian $p$-groups of maximal class.

math.GR↗
arXiv2025

Finite groups with dense ${\cal CD}$-subgroups

A group $G$ is said to have dense ${\cal CD}$-subgroups if each non-empty open interval of the subgroup lattice $L(G)$ contains a subgroup in the Chermak--Delgado lattice ${\cal CD}(G)$. In this note, we study finite groups satisfying this property.

math.GR↗
arXiv2025

Finite groups whose subgroup graph contains a vertex of large degree

T.C. Burness and S.D. Scott \cite{3} classified finite groups $G$ such that the number of prime order subgroups of $G$ is greater than $|G|/2-1$. In this note, we study finite groups $G$ whose subgroup graph contains a vertex of degree greater than $|G|/2-1$. The classification given for finite solvable groups extends the work of Burness and Scott.

math.GR↗
arXiv2024

A remark on the number of automorphisms of finite abelian groups

Let $Ab_0$ be the class of finite abelian groups and consider the function $f:Ab_0\longrightarrow(0,\infty)$ given by $f(G)=\frac{|{\rm Aut}(G)|}{|G|}$\,, where ${\rm Aut}(G)$ is the automorphism group of a finite abelian group $G$. In this short note, we prove that the image of $f$ is a dense set in $[0,\infty)$.

math.GR↗
arXiv2024

Finite groups with dense solitary subgroups

A group $G$ is said to have dense solitary subgroups if each non-empty open interval in its subgroup lattice $L(G)$ contains a solitary subgroup. In this short note, we find all finite groups satisfying this property.

math.GR↗
arXiv2024

On the solvability of a finite group by the sum of subgroup orders

Let $G$ be a finite group and $σ_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. Under some restrictions on the number of conjugacy classes of (non-normal) maximal subgroups of $G$, we prove that if $σ_1(G)<\frac{117}{20}\,$, then $G$ is solvable. This partially solves an open problem posed in \cite{9}.

math.GR↗
arXiv2024

Almost abelian numbers

In this article we introduce the concept of almost $\mathcal{P}$-numbers. We survey the existing results in literature for almost cyclic numbers and give characterizations for almost abelian and almost nilpotent numbers proving these two are equivalent.

math.GR↗