scholarly journals Finite groups whose element orders are consecutive integers

1991 ◽  
Vol 143 (2) ◽  
pp. 388-400 ◽  
Author(s):  
Rolf Brandl ◽  
Shi Wujie
Keyword(s):  
Finite Groups ◽  
Element Orders ◽  
Consecutive Integers

scholarly journals Arithmetic progressions in finite groups

2021 ◽  
pp. 2250096
Author(s):  
Marius Tărnăuceanu
Keyword(s):  
Arithmetic Progression ◽  
Finite Groups ◽  
Abelian Subgroup ◽  
Proper Subgroup ◽  
Arithmetic Progressions ◽  
Element Orders ◽  
Consecutive Integers

In this paper, we describe the structure of finite groups whose element orders or proper (abelian) subgroup orders form an arithmetic progression of ratio [Formula: see text]. This extends the case [Formula: see text] studied in previous papers [R. Brandl and W. Shi, Finite groups whose element orders are consecutive integers, J. Algebra 143 (1991) 388–400; Y. Feng, Finite groups whose abelian subgroup orders are consecutive integers, J. Math. Res. Exp. 18 (1998) 503–506; W. Shi, Finite groups whose proper subgroup orders are consecutive integers, J. Math. Res. Exp. 14 (1994) 165–166].

Variations on results on orders of products in finite groups

2020 ◽  
pp. 1-7
Author(s):  
Juan Martínez ◽  
Alexander Moretó
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Simple Groups ◽  
Prime Power Order ◽  
Finite Simple Groups ◽  
Prime Divisors ◽  
Element Orders ◽  
Hall Subgroups ◽  
A Finite Group

In 2014, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if o(xy) = o(x)o(y) for every x, y ∈ G with (o(x), o(y)) = 1. This has led to a number of results that characterize the nilpotence of a group (or the existence of nilpotent Hall subgroups, or the existence of normal Hall subgroups) in terms of prime divisors of element orders. Here, we look at these results with a new twist. The first of our main results asserts that G is nilpotent if and only if o(xy) ⩽ o(x)o(y) for every x, y ∈ G of prime power order with (o(x), o(y)) = 1. As an immediate consequence, we recover the Baumslag–Wiegold theorem. The proof of this result is elementary. We prove some variations of this result that depend on the classification of finite simple 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].

Sums of Element Orders in Finite Groups

2009 ◽  
Vol 37 (9) ◽  
pp. 2978-2980 ◽  
Author(s):  
Habib Amiri ◽  
S. M. Jafarian Amiri ◽  
I. M. Isaacs
Keyword(s):  
Finite Groups ◽  
Element Orders

Second maximum sum of element orders on finite groups

2014 ◽  
Vol 218 (3) ◽  
pp. 531-539 ◽  
Author(s):  
S.M. Jafarian Amiri ◽  
Mohsen Amiri
Keyword(s):  
Finite Groups ◽  
Element Orders

scholarly journals On the structure of finite groups isospectral to finite simple groups

2015 ◽  
Vol 18 (5) ◽  
Author(s):  
Mariya A. Grechkoseeva ◽  
Andrey V. Vasil'ev
Keyword(s):  
Simple Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Element Orders ◽  
Nonabelian Simple Group

AbstractFinite groups are said to be isospectral if they have the same sets of element orders. A finite nonabelian simple group

scholarly journals Element orders and Sylow structure of finite groups

2005 ◽  
Vol 252 (1) ◽  
pp. 223-230 ◽  
Author(s):  
Gunter Malle ◽  
Alexander Moretó ◽  
Gabriel Navarro
Keyword(s):  
Finite Groups ◽  
Element Orders

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.

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