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 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

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.

Coleman Automorphisms of Extensions of Finite Characteristically Simple Groups by Some Finite Groups

2021 ◽  
Vol 28 (04) ◽  
pp. 561-568
Author(s):  
Jinke Hai ◽  
Lele Zhao
Keyword(s):  
Abelian Group ◽  
Simple Group ◽  
Finite Groups ◽  
Finite Simple Group ◽  
Simple Groups ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Inner Automorphism ◽  
Coleman Automorphism

Let [Formula: see text] be an extension of a finite characteristically simple group by an abelian group or a finite simple group. It is shown that every Coleman automorphism of [Formula: see text] is an inner automorphism. Interest in such automorphisms arises from the study of the normalizer problem for integral group rings.

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.

Mixing, Communication Complexity and Conjectures of Gowers and Viola

2016 ◽  
Vol 26 (4) ◽  
pp. 628-640 ◽  
Author(s):  
ANER SHALEV
Keyword(s):  
Simple Group ◽  
Lower Bounds ◽  
Finite Simple Group ◽  
Decision Problem ◽  
Communication Complexity ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Quantitative Estimates ◽  
Complexity Lower Bounds ◽  
Bounded Rank

We study the distribution of products of conjugacy classes in finite simple groups, obtaining effective two-step mixing results, which give rise to an approximation to a conjecture of Thompson.Our results, combined with work of Gowers and Viola, also lead to the solution of recent conjectures they posed on interleaved products and related complexity lower bounds, extending their work on the groups SL(2,q) to all (non-abelian) finite simple groups.In particular it follows that, ifGis a finite simple group, andA,B⊆Gtfort⩾ 2 are subsets of fixed positive densities, then, asa= (a1, . . .,at) ∈Aandb= (b1, . . .,bt) ∈Bare chosen uniformly, the interleaved producta•b:=a1b1. . .atbtis almost uniform onG(with quantitative estimates) with respect to the ℓ∞-norm.It also follows that the communication complexity of an old decision problem related to interleaved products ofa,b∈Gtis at least Ω(tlog |G|) whenGis a finite simple group of Lie type of bounded rank, and at least Ω(tlog log |G|) whenGis any finite simple group. Both these bounds are best possible.

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 Mathieu group M23

1971 ◽  
Vol 12 (4) ◽  
pp. 385-392 ◽  
Author(s):  
N. Bryce
Keyword(s):  
Simple Group ◽  
Finite Simple Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Unique Determination ◽  
Mathieu Group ◽  
Mathieu Groups

Until 1965, when Janko [7] established the existence of his finite simple group J1, the five Mathieu groups were the only known examples of isolated finite simple groups. In 1951, R. G. Stanton [10] showed that M12 and M24 were determined uniquely by their order. Recent characterizations of M22 and M23 by Janko [8], M22 by D. Held [6], and M11 by W. J. Wong [12], have facilitated the unique determination of the three remaining Mathieu groups by their orders. D. Parrott [9] has so characterized M22 and M11, while this paper is an outline of the characterization of M23 in terms of its order.

Two-variable laws for a class of finite simple groups

1974 ◽  
Vol 10 (1) ◽  
pp. 85-89 ◽  
Author(s):  
Bruce Southcott
Keyword(s):  
Simple Group ◽  
Finite Simple Group ◽  
Simple Groups ◽  
Finite Simple Groups

This paper presents a two-variable basis for the variety generated by the finite simple group PSL(2, 2n).

Boolean simple groups and boolean simple rings

1988 ◽  
Vol 53 (1) ◽  
pp. 160-173
Author(s):  
Gaisi Takeuti
Keyword(s):  
Boolean Algebra ◽  
Simple Group ◽  
Set Theory ◽  
Finite Simple Group ◽  
Complete Boolean Algebra ◽  
Simple Groups ◽  
Finite Simple Groups

Let be a complete Boolean algebra and G a finite simple group in the Scott-Solovay -valued model V() of set theory. If we observe G outside V(), then we get a new group which is denoted by Ĝ. In general, Ĝ is not finite nor simple. Nevertheless Ĝ satisfies every property satisfied by a finite simple group with some translation. In this way, we can get a class of groups for which we can use a well-developed theory of the finite simple groups. We call Ĝ Boolean simple if G is simple in some V(). In the same way we define Boolean simple rings. The main purpose of this paper is a study of structures of Boolean simple groups and Boolean simple rings. As for Boolean simple rings, K. Eda previously constructed Boolean completion of rings with a certain condition. His construction is useful for our purpose.The present work is a part of a series of systematic applications of Boolean valued method. The reader who is interested in this subject should consult with papers by Eda, Nishimura, Ozawa, and the author in the list of references.

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

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