scholarly journals Large element orders and the characteristic of Lie-type simple groups

2009 ◽  
Vol 322 (3) ◽  
pp. 802-832 ◽  
Author(s):  
William M. Kantor ◽  
Ákos Seress
Keyword(s):  
Simple Groups ◽  
Large Element ◽  
Element Orders

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.

On Constructive Recognition of Finite Simple Groups by Element Orders

2014 ◽  
Vol 53 (4) ◽  
pp. 349-351 ◽  
Author(s):  
A. A. Buturlakin ◽  
A. V. Vasil’ev
Keyword(s):  
Simple Groups ◽  
Finite Simple 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

RECOGNITION OF SOME FINITE SIMPLE GROUPS OF TYPE Dn(q) BY SPECTRUM

2009 ◽  
Vol 19 (05) ◽  
pp. 681-698 ◽  
Author(s):  
HUAIYU HE ◽  
WUJIE SHI
Keyword(s):  
Finite Group ◽  
Finite Simple Group ◽  
Composition Factor ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Element Orders ◽  
Set Of Element Orders ◽  
A Finite Group ◽  
Group D ◽  
Nonabelian Composition Factor

The spectrum ω(G) of a finite group G is the set of element orders of G. Let L be finite simple group Dn(q) with disconnected Gruenberg–Kegel graph. First, we establish that L is quasi-recognizable by spectrum except D4(2) and D4(3), i.e., every finite group G with ω(G) = ω(L) has a unique nonabelian composition factor that is isomorphic to L. Second, for some special series of integers n, we prove that L is recognizable by spectrum, i.e., every finite group G with ω(G) = ω(L) is isomorphic to L.

Element Orders in Covers of Finite Simple Groups

2013 ◽  
Vol 52 (5) ◽  
pp. 426-428 ◽  
Author(s):  
M. A. Grechkoseeva
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups ◽  
Element Orders

Recognition of simple groups B p (3) by the set of element orders

2010 ◽  
Vol 51 (2) ◽  
pp. 244-254 ◽  
Author(s):  
Rulin Shen ◽  
Wujie Shi ◽  
M. R. Zinov’eva
Keyword(s):  
Simple Groups ◽  
Element Orders ◽  
Set Of Element Orders

scholarly journals Recognition of the Simple Groups 2D8((2n)2)

2019 ◽  
Vol 57 (2) ◽  
pp. 53-60
Author(s):  
Behnam Ebrahimzadeh
Keyword(s):  
Prime Number ◽  
Finite Groups ◽  
Specific Property ◽  
Element Order ◽  
Simple Groups ◽  
Element Orders

Abstract One of the important problems in finite groups theory is group characterization by specific property. Properties, such as element orders, set of elements with the same order, the largest element order, etc. In this paper, we prove that the simple groups 2 D 8((2 n )2)where, 28 n + 1 is a prime number are uniquely determined by its order and the largest elements order.

ON THE SUM OF ELEMENT ORDERS OF FINITE SIMPLE GROUPS

2013 ◽  
Vol 12 (07) ◽  
pp. 1350026 ◽  
Author(s):  
YADOLLAH MAREFAT ◽  
ALI IRANMANESH ◽  
ABOLFAZL TEHRANIAN
Keyword(s):  
Finite Group ◽  
Simple Groups ◽  
Short Paper ◽  
Finite Simple Groups ◽  
Element Orders ◽  
A Finite Group

Let G be a finite group and ψ(G) = ∑g∈Go(g), where o(g) denotes the order of g ∈ G. In this short paper we show that the conjecture of minimality of ψ(G) in simple groups, posed in [J. Algebra Appl. 10(2) (2011) 187–190], is incorrect.

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