arXiv subjects

Marius Tarnauceanu

Publications and source records attributed to Marius Tarnauceanu.

17 recordsLinked to original sources

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↗

arXiv2016

Finite groups determined by an inequality of the orders of their subgroups II

In this note we study a class of finite groups for which the orders of subgroups satisfy a certain inequality. In particular, characterizations of the well-known groups $\mathbb{Z}_2\times\mathbb{Z}_2$ and $S_3$ are obtained.

math.GR↗

arXiv2016

Cyclic subgroup commutativity degrees of finite groups

In this paper we introduce and study the concept of cyclic subgroup commutativity degree of a finite group $G$. This quantity measures the probability of two random cyclic subgroups of $G$ commuting. Explicit formulas are obtained for some particular classes of groups. A criterion for a finite group to be an Iwasawa group is also presented.

math.GR↗

arXiv2016

On a generalization of the Gauss's formula

In this paper we study a group theoretical generalization of the well-known Gauss's formula that uses the generalized Euler's totient function introduced in [11].

math.GR↗

arXiv2016

A characterization of PSL(2,q), q = 5,7

In this short note we prove that the finite non-abelian simple groups PSL(2,q), where q = 5,7, are determined by their posets of classes of isomorphic subgroups. In particular, this disproves the conjecture in the end of [5].

math.GR↗

arXiv2015

The normal subgroup structure of ZM-groups

The main goal of this note is to determine and to count the normal subgroups of a ZM-group. We also indicate some necessary and sufficient conditions such that the normal subgroups of a ZM-group form a chain.

math.GR↗

arXiv2015

Non-CLT groups of order $pq^3$

In this note we give a characterization of finite groups of order $pq^3$ ($p$, $q$ primes) that fail to satisfy the Converse of Lagrange's Theorem.

math.GR↗

arXiv2015

On the factorization numbers of some finite $p$-groups

This note deals with the computation of the factorization number $F_2(G)$ of a finite group $G$. By using the Möbius inversion formula, explicit expressions of $F_2(G)$ are obtained for two classes of finite abelian groups, improving the results of {\it Factorization numbers of some finite groups}, Glasgow Math. J. (2012).

math.GR↗

arXiv2015

On the converse of Fuzzy Lagrange's Theorem

In fuzzy group theory many versions of the well-known Lagrange's theorem have been studied. The aim of this article is to investigate the converse of one of those results. This leads to an interesting characterization of finite cyclic groups.

math.GR↗

arXiv2015

On finite groups with dismantlable subgroup lattices

In this note we study the finite groups whose subgroup lattices are dismantlable.

math.GR↗

arXiv2015

The posets of classes of isomorphic subgroups of finite groups

In this paper we introduce and study the poset of equivalence classes of subgroups of a finite group $G$, induced by the isomorphism relation. This contains the well-known lattice of solitary subgroups of $G$. We prove that in several particular cases it determines the structure of $G$.

math.GR↗

arXiv2015

Finite groups with a certain number of cyclic subgroups

In this short note, we describe the finite groups $G$ having $|G|-1$ cyclic subgroups. This leads to a nice characterization of the symmetric group $S_3$.

math.HO↗

arXiv2013

Normality degrees of finite groups

In this paper we introduce and study the concept of normality degree of a finite group $G$. This quantity measures the probability of a random subgroup of $G$ to be normal. Explicit formulas are obtained for some particular classes of finite groups. Several limits of normality degrees are also computed.

math.GR↗

arXiv2013

A generalization of the Euler's totient function

The main goal of this paper is to provide a group theoretical generalization of the well-known Euler's totient function. This determines an interesting class of finite groups.

math.GR↗

arXiv2013

On the number of diamonds in the subgroup lattice of a finite abelian group

The main goal of the current paper is to determine the total number of diamonds in the subgroup lattice of a finite abelian group. This counting problem is reduced to finite $p$-groups. Explicit formulas are obtained in some particular cases.

math.GR↗

arXiv2013

The subgroup commutativity degree of finite P-groups

The subgroup commutativity degree of a group G has been defined in [6] as the probability that two subgroups of G commute, or equivalently that the product of two subgroups is again a subgroup. Problem 4.3 of [6] asks whether there exist families of groups other than dihedral, quasi-dihedral or generalized quaternion (all of 2-power cardinality), whose subgroup commutativity degree tends to 0 as the size of the group tends to infinity. An affirmative answer to this question has been provided by Aivazidis [1, 2] for the family of projective special linear groups over fields of even characteristic and for the family of the simple Suzuki groups. In this short note we indicate another family of groups with this property, namely the finite P-groups.

math.GR↗

arXiv2012

A generalization of Menon's identity

In this note we give a generalization of the well-known Menon's identity. This is based on applying the Burnside's lemma to a certain group action.

math.GR↗