The lattice properties of 𝔛-local formations of finite groups

2020 ◽  
pp. 2150043 ◽  
Author(s):  
Aleksandr Tsarev
Keyword(s):  
Finite Groups ◽  
Simple Groups ◽  
Local Formation ◽  
Common Extension ◽  
Completeness Property

Let [Formula: see text] be a class of simple groups with a completeness property [Formula: see text]. Förster introduced the concept of [Formula: see text]-local formation in order to obtain a common extension of well-known theorems of Gaschütz–Lubeseder–Schmid and Baer [Publ. Mat. UAB 29(2–3) (1985) 39–76]. In this paper, it is proved that the lattice of all [Formula: see text]-local formations of finite groups is modular.

scholarly journals Frattini classes of saturated formations of finite groups

1990 ◽  
Vol 42 (2) ◽  
pp. 267-286 ◽  
Author(s):  
Peter Förster
Keyword(s):  
Finite Groups ◽  
Local Formation ◽  
Formation Of Finite Groups ◽  
Saturated Formations ◽  
Definition Of

We study the following question: given any local formation of finite groups, do there exist maximal local subformations? An answer is given by providing a local definition of the intersection of all maximal local subformations.

scholarly journals Finite groups acting on hyperelliptic 3-manifolds

2020 ◽  
Vol 29 (04) ◽  
pp. 2050021
Author(s):  
Mattia Mecchia
Keyword(s):  
Fixed Point ◽  
Simple Group ◽  
Finite Groups ◽  
Prime Power ◽  
Simple Groups ◽  
Fixed Point Set ◽  
Point Set

We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to [Formula: see text] Such involutions are called hyperelliptic as the manifolds admitting such an action. We consider finite groups acting on 3-manifolds and containing hyperelliptic involutions whose fixed-point set has [Formula: see text] components. In particular we prove that a simple group containing such an involution is isomorphic to [Formula: see text] for some odd prime power [Formula: see text], or to one of four other small simple groups.

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 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).

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.

scholarly journals Low-Dimensional Representations of Quasi-Simple Groups

2001 ◽  
Vol 4 ◽  
pp. 22-63 ◽  
Author(s):  
Gerhard Hiss ◽  
Gunter Malle
Keyword(s):  
Finite Groups ◽  
Irreducible Representations ◽  
Simple Groups ◽  
Additional Information ◽  
Brauer Character ◽  
Low Dimensional

AbstractThe authors determine all the absolutely irreducible representations of degree up to 250 of quasi-simple finite groups, excluding groups that are of Lie type in their defining characteristic. Additional information is also given on the Frobenius-Schur indicators and the Brauer character fields of the representations.

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.

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