The group J4 × J4 is recognizable by spectrum

2020 ◽  
pp. 2150061
Author(s):  
Ilya B. Gorshkov ◽  
Natalia V. Maslova
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Finite Groups ◽  
Element Orders ◽  
Sporadic Group ◽  
A Finite Group

The spectrum of a finite group is the set of its element orders. In this paper, we prove that the direct product of two copies of the finite simple sporadic group [Formula: see text] is uniquely determined by its spectrum in the class of all finite groups.

PROPERTIES OF FINITE GROUPS DETERMINED BY THE PRODUCT OF THEIR ELEMENT ORDERS

2020 ◽  
pp. 1-8
Author(s):  
MORTEZA BANIASAD AZAD ◽  
BEHROOZ KHOSRAVI
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Element Orders ◽  
A Finite Group

For a finite group $G$ , define $l(G)=(\prod _{g\in G}o(g))^{1/|G|}/|G|$ , where $o(g)$ denotes the order of $g\in G$ . We prove that if $l(G)>l(A_{5}),l(G)>l(A_{4}),l(G)>l(S_{3}),l(G)>l(Q_{8})$ or $l(G)>l(C_{2}\times C_{2})$ , then $G$ is solvable, supersolvable, nilpotent, abelian or cyclic, respectively.

An answer to a conjecture on the sum of element orders

2020 ◽  
pp. 2250067
Author(s):  
Morteza Baniasad Azad ◽  
Behrooz Khosravi ◽  
Morteza Jafarpour
Keyword(s):  
Finite Group ◽  
Lower Bound ◽  
Nilpotent Group ◽  
Prime Number ◽  
Lower Bounds ◽  
Finite Groups ◽  
Open Problem ◽  
Element Orders ◽  
Equality Condition ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text], where [Formula: see text] denotes the order of [Formula: see text]. The function [Formula: see text] was introduced by Tărnăuceanu. In [M. Tărnăuceanu, Detecting structural properties of finite groups by the sum of element orders, Israel J. Math. (2020), https://doi.org/10.1007/s11856-020-2033-9 ], some lower bounds for [Formula: see text] are determined such that if [Formula: see text] is greater than each of them, then [Formula: see text] is cyclic, abelian, nilpotent, supersolvable and solvable. Also, an open problem aroused about finite groups [Formula: see text] such that [Formula: see text] is equal to the amount of each lower bound. In this paper, we give an answer to the equality condition which is a partial answer to the open problem posed by Tărnăuceanu. Also, in [M. Baniasad Azad and B. Khosravi, A criterion for p-nilpotency and p-closedness by the sum of element orders, Commun. Algebra (2020), https://doi.org/10.1080/00927872.2020.1788571 ], it is shown that: If [Formula: see text], where [Formula: see text] is a prime number, then [Formula: see text] and [Formula: see text] is cyclic. As the next result, we show that if [Formula: see text] is not a [Formula: see text]-nilpotent group and [Formula: see text], then [Formula: see text].

Blocks of defect zero in finite groups with conjugate subgroups of a given prime order

2017 ◽  
Vol 16 (11) ◽  
pp. 1750217
Author(s):  
Tianze Li ◽  
Yanjun Liu ◽  
Guohua Qian
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Direct Product ◽  
Finite Groups ◽  
Prime Order ◽  
Text Block ◽  
Order Group ◽  
Odd Order ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] be a prime. In this note, we show that if [Formula: see text] and all subgroups of [Formula: see text] of order [Formula: see text] are conjugate, then either [Formula: see text] has a [Formula: see text]-block of defect zero, or [Formula: see text] and [Formula: see text] is a direct product of a simple group [Formula: see text] and an odd order group. This improves one of our previous works.

scholarly journals Algebraic Properties of Implication-Based Anti-Fuzzy Subgroup over a Finite Group

2019 ◽  
Vol 9 (1S4) ◽  
pp. 1116-1120
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Finite Groups ◽  
Algebraic Properties ◽  
Fuzzy Subgroup ◽  
Fuzzy Subgroups ◽  
A Finite Group ◽  
Fuzzy Direct Product

This article deals with few algebraic characteristics of implication-based anti-fuzzy subgroup of a finite group.In addition, the implication-based anti-fuzzy direct product of implication-based anti-fuzzy subgroups over finite groups is developed and studied elaborately. The condition for an implication-based anti-fuzzy subgroup of a finite group to be a conjugate to another implication-based anti-fuzzy subgroup is conceptualized. Some of their characteristics are investigated in this paper.

Product of Implication-Based Intuitionistic Fuzzy Semiautomatons over Finite Groups

2019 ◽  
Vol 15 (03) ◽  
pp. 503-515 ◽  
Author(s):  
M. Selvarathi
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Direct Product ◽  
Finite Groups ◽  
Fundamental Properties ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Subgroup ◽  
Fuzzy Subgroups ◽  
A Finite Group

In this paper, Implication-based intuitionistic fuzzy semiautomaton (IB-IFSA) of a finite group is defined and investigated. The theory of an implication-based intuitionistic fuzzy kernel and implication-based intuitionistic fuzzy subsemiautomaton of an IB-IFSA over a finite group are formulated using the approach of implication-based intuitionistic fuzzy subgroup and implication-based intuitionistic fuzzy normal subgroup. The product of implication-based intuitionistic fuzzy subgroups is postulated and investigated. Further, direct product of implication-based intuitionistic fuzzy semiautomatons over the finite groups is elaborately studied. Fundamental properties concerning them are also dealt with.

Finite groups having unique proper characteristic subgroups. I

1955 ◽  
Vol 51 (1) ◽  
pp. 25-36 ◽  
Author(s):  
D. R. Taunt
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Finite Groups ◽  
Simple Groups ◽  
Characteristic Subgroup ◽  
Direct Products ◽  
A Finite Group

It is well known that a characteristically-simple finite group, that is, a group having no characteristic subgroup other than itself and the identity subgroup, must be either simple or the direct product of a number of isomorphic simple groups. It was suggested to the author by Prof. Hall that finite groups possessing exactly one proper characteristic subgroup would repay attention. We shall call a finite group having a unique proper characteristic subgroup a ‘UCS group’. In the present paper we first give some results on direct products of isomorphic UCS groups, and then we consider in more detail one of the types of UCS groups which can exist, that consisting of groups whose orders are divisible by exactly two distinct primes.

SUM OF ELEMENT ORDERS ON FINITE GROUPS OF THE SAME ORDER

2011 ◽  
Vol 10 (02) ◽  
pp. 187-190 ◽  
Author(s):  
H. AMIRI ◽  
S. M. JAFARIAN AMIRI
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Cyclic Group ◽  
Finite Groups ◽  
Element Orders ◽  
Minimum Value ◽  
Maximum Value ◽  
A Finite Group

For a finite group G, let ψ(G) denote the sum of element orders of G. It is known that the maximum value of ψ on the set of groups of order n, where n is a positive integer, will occur at the cyclic group Cn. In this paper, we investigate the minimum value of ψ on the set of groups of the same order.

scholarly journals c-SECTIONS, SOLVABILITY AND LARGE SUBGROUPS OF FINITE GROUPS

2012 ◽  
Vol 86 (2) ◽  
pp. 291-302
Author(s):  
BARBARA BAUMEISTER ◽  
GIL KAPLAN
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Existence Result ◽  
Composition Factor ◽  
Maximal Subgroups ◽  
Fundamental Result ◽  
Equivalent Condition ◽  
Sporadic Group ◽  
A Finite Group

Abstractc-Sections of maximal subgroups in a finite group and their relation to solvability have been extensively researched in recent years. A fundamental result due to Wang [‘C-normality of groups and its properties’, J. Algebra 180 (1998), 954–965] is that a finite group is solvable if and only if the c-sections of all its maximal subgroups are trivial. In this paper we prove that if for each maximal subgroup of a finite group G, the corresponding c-section order is smaller than the index of the maximal subgroup, then each composition factor of G is either cyclic or isomorphic to the O’Nan sporadic group (the converse does not hold). Furthermore, by a certain ‘refining’ of the latter theorem we obtain an equivalent condition for solvability. Finally, we provide an existence result for large subgroups in the sense of Lev [‘On large subgroups of finite groups’ J. Algebra 152 (1992), 434–438].

scholarly journals Finite groups with a trivial Chermak–Delgado subgroup

2018 ◽  
Vol 21 (3) ◽  
pp. 449-461 ◽  
Author(s):  
Ryan McCulloch
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Finite Groups ◽  
Extension Theorem ◽  
Cartesian Product ◽  
Direct Factor ◽  
Minimal Group ◽  
Minimal Groups ◽  
A Finite Group ◽  
Lattice Of Subgroups

Abstract The Chermak–Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups of G. The least element of the Chermak–Delgado lattice of G is known as the Chermak–Delgado subgroup of G. This paper concerns groups with a trivial Chermak–Delgado subgroup. We prove that if the Chermak–Delgado lattice of such a group is lattice isomorphic to a Cartesian product of lattices, then the group splits as a direct product, with the Chermak–Delgado lattice of each direct factor being lattice isomorphic to one of the lattices in the Cartesian product. We establish many properties of such groups and properties of subgroups in the Chermak–Delgado lattice. We define a CD-minimal group to be an indecomposable group with a trivial Chermak–Delgado subgroup. We establish lattice theoretic properties of Chermak–Delgado lattices of CD-minimal groups. We prove an extension theorem for CD-minimal groups, and use the theorem to produce twelve examples of CD-minimal groups, each having different CD lattices. Curiously, quasi-antichain p-group lattices play a major role in the author’s constructions.

scholarly journals Bounds on the Action Degree of Groups

MATEMATIKA ◽  
2019 ◽  
Vol 35 (2) ◽  
pp. 271-282
Author(s):  
S. Alrehaili ◽  
Charef Beddani
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Finite Groups ◽  
Random Element ◽  
General Relation ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
A Finite Group

The commutativity degree is the probability that a pair of elements chosen randomly from a group commute. The concept of  commutativity degree has been widely discussed by several authors in many directions.  One of the important generalizations of commutativity degree is the probability that a random element from a finite group G fixes a random element from a non-empty set S that we call the action degree of groups. In this research, the concept of action degree is further studied where some inequalities and bounds on the action degree of finite groups are determined.  Moreover, a general relation between the action degree of a finite group G and a subgroup H is provided. Next, the action degree for the direct product of two finite groups is determined. Previously, the action degree was only defined for finite groups, the action degree for finitely generated groups will be defined in this research and some bounds on them are going to be determined.

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