scholarly journals On the laws of certain finite groups

1970 ◽  
Vol 11 (4) ◽  
pp. 441-489 ◽  
Author(s):  
John Cossey ◽  
Sheila Oates MacDonald ◽  
Anne Penfold Street
Keyword(s):  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Classification Problems ◽  
Sylow Subgroups ◽  
Local Finiteness ◽  
Special Cases

In recent years a great deal of attention has been devoted to the study of finite simple groups, but one aspect which seems to have been little considered is that of the laws they satisfy. In a recent paper [3], the first two of the present authors gave a basis for laws of PSL(2, 5). The techniques of [3] can be used to show that (modulo certain classification problems) a basis for the laws of PSL(2, pn) can be made up from laws of the following types:(1) an exponent law,(2) laws which determine the Sylow subgroups,(3) laws which determine the normalisers of the Sylow subgroups,(4) in certain special cases, laws which determine subvarieties of smaller exponent, e.g. the subvariety of exponent 12 for those PSL(2, pn) which contain S4,(5) a law implying local finiteness.

On the number of Sylow subgroups in finite simple groups

2020 ◽  
pp. 2150115
Author(s):  
Zhenfeng Wu
Keyword(s):  
Group Theory ◽  
Simple Group ◽  
Finite Groups ◽  
Finite Simple Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Sylow Subgroups

Denote by [Formula: see text] the number of Sylow [Formula: see text]-subgroups of [Formula: see text]. For every subgroup [Formula: see text] of [Formula: see text], it is easy to see that [Formula: see text], but [Formula: see text] does not divide [Formula: see text] in general. Following [W. Guo and E. P. Vdovin, Number of Sylow subgroups in finite groups, J. Group Theory 21(4) (2018) 695–712], we say that a group [Formula: see text] satisfies DivSyl(p) if [Formula: see text] divides [Formula: see text] for every subgroup [Formula: see text] of [Formula: see text]. In this paper, we show that “almost for every” finite simple group [Formula: see text], there exists a prime [Formula: see text] such that [Formula: see text] does not satisfy DivSyl(p).

scholarly journals A Study About One Generation of Finite Simple Groups and Finite Groups

2021 ◽  
Vol 13 (3) ◽  
pp. 59
Author(s):  
Nader Taffach
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Prime Numbers ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
A Finite Group

In this paper, we study the problem of how a finite group can be generated by some subgroups. In order to the finite simple groups, we show that any finite non-abelian simple group can be generated by two Sylow p1 - and p_2 -subgroups, where p_1  and p_2  are two different primes. We also show that for a given different prime numbers p  and q , any finite group can be generated by a Sylow p -subgroup and a q -subgroup.

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.

scholarly journals Number of Sylow subgroups in finite groups

2018 ◽  
Vol 21 (4) ◽  
pp. 695-712 ◽  
Author(s):  
Wenbin Guo ◽  
Evgeny P. Vdovin
Keyword(s):  
Finite Groups ◽  
Simple Groups ◽  
Alternative Proof ◽  
Sylow Subgroups ◽  
Almost Simple Groups

AbstractDenote by {\nu_{p}(G)} the number of Sylow p-subgroups of G. It is not difficult to see that {\nu_{p}(H)\leqslant\nu_{p}(G)} for {H\leqslant G}, however {\nu_{p}(H)} does not divide {\nu_{p}(G)} in general. In this paper we reduce the question whether {\nu_{p}(H)} divides {\nu_{p}(G)} for every {H\leqslant G} to almost simple groups. This result substantially generalizes the previous result by G. Navarro and also provides an alternative proof of Navarro’s theorem.

On mutually permutable products of generalized supersoluble subgroups of finite groups

2016 ◽  
Vol 09 (03) ◽  
pp. 1650054
Author(s):  
E. N. Myslovets
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Paper Properties ◽  
Supersoluble Groups ◽  
A Finite Group ◽  
Mutually Permutable Products ◽  
Composition Factors

Let [Formula: see text] be a class of finite simple groups. We say that a finite group [Formula: see text] is a [Formula: see text]-group if all composition factors of [Formula: see text] are contained in [Formula: see text]. A group [Formula: see text] is called [Formula: see text]-supersoluble if every chief [Formula: see text]-factor of [Formula: see text] is a simple group. In this paper, properties of mutually permutable products of [Formula: see text]-supersoluble finite groups are studied. Some earlier results on mutually permutable products of [Formula: see text]-supersoluble groups (SC-groups) appear as particular cases.

Another Existence and Uniqueness Proof of the Tits Group

2008 ◽  
Vol 15 (02) ◽  
pp. 241-278
Author(s):  
Gerhard O. Michler ◽  
Lizhong Wang
Keyword(s):  
Finite Groups ◽  
Existence And Uniqueness ◽  
Frobenius Group ◽  
Nilpotency Class ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Uniqueness Proof ◽  
Groups Of Lie Type ◽  
Uniqueness Criteria ◽  
Tits Group

In this article we give a self-contained existence and uniqueness proof for the Tits simple group T. Parrott gave the first uniqueness proof. Whereas Tits' and Parrott's results employ the theory of finite groups of Lie type, our existence and uniqueness proof follows from the general algorithms and uniqueness criteria for abstract finite simple groups described in the first author's book [11]. All we need from the previous papers is the fact that the centralizer H of the Tits group T is an extension of a 2-group J with order 29 and nilpotency class 3 by a Frobenius group F of order 20 such that the center Z(H) has order 2 and any Sylow 5-subgroup Q of H has a centralizer CJ(Q) ≤ Z(H).

A Class of Finite Groups with Abelian Sylow p-Subgroups

2006 ◽  
Vol 13 (03) ◽  
pp. 471-480
Author(s):  
Zhikai Zhang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Defect Groups ◽  
Abelian Defect Groups ◽  
A Finite Group

In this paper, we first determine the structure of the Sylow p-subgroup P of a finite group G containing no elements of order 2p (p > 2), and then show that the Broué Abelian Defect Groups Conjecture is true for the principal p-block of G. The result depends on the classification of finite simple groups.

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 Minimal length factorizations of finite simple groups of Lie type by unipotent Sylow subgroups

2016 ◽  
Vol 19 (2) ◽  
Author(s):  
Martino Garonzi ◽  
Dan Levy ◽  
Attila Maróti ◽  
Iulian I. Simion
Keyword(s):  
Simple Group ◽  
Finite Simple Group ◽  
Minimal Length ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Sylow Subgroups ◽  
Groups Of Lie Type

AbstractWe prove that every finite simple group

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