Generating Pairs for the Fischer Group Fi23

2020 ◽  
Vol 27 (04) ◽  
pp. 713-730
Author(s):  
Faryad Ali ◽  
Mohammed Al-Kadhi
Keyword(s):  
Simple Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Prime Divisors ◽  
Sporadic Simple Group ◽  
Triangle Group ◽  
Fischer Group

A group G is said to be (l, m, n)-generated if it is a quotient of the triangle group [Formula: see text]. Moori posed in 1993 the question of finding all the triples (l, m, n) such that non-abelian finite simple groups are (l, m, n)-generated. We partially answer this question for the Fischer sporadic simple group Fi23. In particular, we investigate all (2, q, r)-generations for the Fischer sporadic simple group Fi23, where q and r are distinct prime divisors of |Fi23|.

scholarly journals Character tables of certain finite simple groups

1971 ◽  
Vol 5 (1) ◽  
pp. 1-42 ◽  
Author(s):  
David C. Hunt
Keyword(s):  
Simple Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Sporadic Simple Group ◽  
Character Tables

M(22) is the sporadic simple group discovered by Fischer in 1969. The character tables of M(22) and the simple groups PSU(5, 2), PSU(6, 2), PSΩ+(6, 3) and PSΩ(7, 3) are presented. Only a brief description of the methods used to determine the tables is given.

scholarly journals Quasirecognition by Prime Graph of the Groups 2D2n(q) Where q < 105

2017 ◽  
Author(s):  
Hossein Moradi ◽  
Mohammad Reza Darafsheh ◽  
Ali Iranmanesh
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Prime Power ◽  
Prime Graph ◽  
Composition Factor ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Prime Divisors ◽  
A Finite Group

Let G be a finite group. The prime graph &Gamma;(G) of G is defined as follows: The set of vertices of&nbsp;&Gamma;(G) is the set of prime divisors of |G| and two distinct vertices p and p' are connected in &Gamma;(G), whenever G has an element of order pp'. A non-abelian simple group P is called recognizable by prime graph if for any finite group G with &Gamma;(G)=&Gamma;(P), G has a composition factor isomorphic to P. In&nbsp;[4] proved finite simple groups 2Dn(q), where&nbsp;n&nbsp;&ne; 4k are quasirecognizable by prime graph. Now in this paper we discuss the quasirecognizability by prime graph of the simple groups 2D2k(q), where&nbsp;k &ge; 9 and q is a prime power less than 105.

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&nbsp; and p_2&nbsp; are two different primes. We also show that for a given different prime numbers p&nbsp; and q , any finite group can be generated by a Sylow p -subgroup and a q -subgroup.

scholarly journals On the Number of p-Regular Elements in Finite Simple Groups

2009 ◽  
Vol 12 ◽  
pp. 82-119 ◽  
Author(s):  
László Babai ◽  
Péter P. Pálfy ◽  
Jan Saxl
Keyword(s):  
Simple Group ◽  
Finite Subgroup ◽  
Simple Groups ◽  
Alternating Group ◽  
Arbitrary Field ◽  
Finite Simple Groups ◽  
Regular Elements ◽  
Exceptional Groups ◽  
Monte Carlo Algorithms ◽  
A Finite Group

AbstractA p-regular element in a finite group is an element of order not divisible by the prime number p. We show that for every prime p and every finite simple group S, a fair proportion of elements of S is p-regular. In particular, we show that the proportion of p-regular elements in a finite classical simple group (not necessarily of characteristic p) is greater than 1/(2n), where n – 1 is the dimension of the projective space on which S acts naturally. Furthermore, in an exceptional group of Lie type this proportion is greater than 1/15. For the alternating group An, this proportion is at least 26/(27√n), and for sporadic simple groups, at least 2/29.We also show that for an arbitrary field F, if the simple group S is a quotient of a finite subgroup of GLn(F) then for any prime p, the proportion of p-regular elements in S is at least min{1/31, 1/(2n)}.Along the way we obtain estimates for the proportion of elements of certain primitive prime divisor orders in exceptional groups, complementing work by Niemeyer and Praeger (1998).Our result shows that in finite simple groups, p-regular elements can be found efficiently by random sampling. This is a key ingredient to recent polynomial-time Monte Carlo algorithms for matrix groups.Finally we complement our lower bound results with the following upper bound: for all n ≥ 2 there exist infinitely many prime powers q such that the proportion of elements of odd order in PSL(n,q) is less than 3/√n.

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 A note on growth sequences of finite simple groups

1995 ◽  
Vol 51 (3) ◽  
pp. 495-499 ◽  
Author(s):  
Ahmad Erfanian ◽  
James Wiegold
Keyword(s):  
Simple Group ◽  
Subgroup Lattice ◽  
Möbius Function ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Mobius Function ◽  
New Formula

The aim of this paper is to give a new precise formula for h(n, A), where A is a finite non-abelian simple group, h(n, A) is the maximum number such that Ah(n, A) can ke generated by n elements, and n ≥ 2. P. Hall gave a formula for h(n, A) in terms of the Möbius function of the subgroup lattice of A; the new formula involves a concept called cospread associated with that of spread as explained in Brenner and Wiegold (1975).

On a conjecture about the quasirecognition by prime graph of some finite simple groups

2019 ◽  
Vol 18 (04) ◽  
pp. 1950070
Author(s):  
Ali Mahmoudifar
Keyword(s):  
Computer Program ◽  
Natural Number ◽  
Simple Group ◽  
Prime Number ◽  
Prime Power ◽  
Prime Graph ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Number Formula ◽  
Finite Linear

It is proved that some finite simple groups are quasirecognizable by prime graph. In [A. Mahmoudifar and B. Khosravi, On quasirecognition by prime graph of the simple groups [Formula: see text] and [Formula: see text], J. Algebra Appl. 14(1) (2015) 12pp], the authors proved that if [Formula: see text] is a prime number and [Formula: see text], then there exists a natural number [Formula: see text] such that for all [Formula: see text], the simple group [Formula: see text] (where [Formula: see text] is a linear or unitary simple group) is quasirecognizable by prime graph. Also[Formula: see text] in that paper[Formula: see text] the author posed the following conjecture: Conjecture. For every prime power [Formula: see text] there exists a natural number [Formula: see text] such that for all [Formula: see text] the simple group [Formula: see text] is quasirecognizable by prime graph. In this paper [Formula: see text] as the main theorem we prove that if [Formula: see text] is a prime power and satisfies some especial conditions [Formula: see text] then there exists a number [Formula: see text] associated to [Formula: see text] such that for all [Formula: see text] the finite linear simple group [Formula: see text] is quasirecognizable by prime graph. Finally [Formula: see text] by a calculation via a computer program [Formula: see text] we conclude that the above conjecture is valid for the simple group [Formula: see text] where [Formula: see text] [Formula: see text] is an odd number and [Formula: see text].

ON A CONJECTURE ON THE PERMUTATION CHARACTERS OF FINITE PRIMITIVE GROUPS

2019 ◽  
Vol 102 (1) ◽  
pp. 77-90
Author(s):  
PABLO SPIGA
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Simple Group ◽  
Simple Groups ◽  
Alternating Group ◽  
Permutation Groups ◽  
Sporadic Simple Group ◽  
Almost Simple Groups ◽  
A Finite Group ◽  
Groups Of Lie Type

Let $G$ be a finite group with two primitive permutation representations on the sets $\unicode[STIX]{x1D6FA}_{1}$ and $\unicode[STIX]{x1D6FA}_{2}$ and let $\unicode[STIX]{x1D70B}_{1}$ and $\unicode[STIX]{x1D70B}_{2}$ be the corresponding permutation characters. We consider the case in which the set of fixed-point-free elements of $G$ on $\unicode[STIX]{x1D6FA}_{1}$ coincides with the set of fixed-point-free elements of $G$ on $\unicode[STIX]{x1D6FA}_{2}$, that is, for every $g\in G$, $\unicode[STIX]{x1D70B}_{1}(g)=0$ if and only if $\unicode[STIX]{x1D70B}_{2}(g)=0$. We have conjectured in Spiga [‘Permutation characters and fixed-point-free elements in permutation groups’, J. Algebra299(1) (2006), 1–7] that under this hypothesis either $\unicode[STIX]{x1D70B}_{1}=\unicode[STIX]{x1D70B}_{2}$ or one of $\unicode[STIX]{x1D70B}_{1}-\unicode[STIX]{x1D70B}_{2}$ and $\unicode[STIX]{x1D70B}_{2}-\unicode[STIX]{x1D70B}_{1}$ is a genuine character. In this paper we give evidence towards the veracity of this conjecture when the socle of $G$ is a sporadic simple group or an alternating group. In particular, the conjecture is reduced to the case of almost simple groups of Lie type.

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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