scholarly journals Centralizers of abelian subgroups in locally finite simple groups

1997 ◽  
Vol 40 (2) ◽  
pp. 217-225
Author(s):  
M. Kuzucuoǧlu
Keyword(s):  
Simple Group ◽  
Finite Length ◽  
Finite Simple Group ◽  
Abelian Subgroup ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Locally Finite ◽  
Odd Order ◽  
Non Linear ◽  
Abelian Subgroups

It is shown that, if a non-linear locally finite simple group is a union of finite simple groups, then the centralizer of every element of odd order has a series of finite length with factors which are either locally solvable or non-abelian simple. Moreover, at least one of the factors is non-linear simple. This is also extended to abelian subgroup of odd orders.

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).

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.

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 CLASSIFICATION OF TETRAVALENT -TRANSITIVE NONNORMAL CAYLEY GRAPHS OF FINITE SIMPLE GROUPS

2021 ◽  
pp. 1-9
Author(s):  
XIN GUI FANG ◽  
JIE WANG ◽  
SANMING ZHOU
Keyword(s):  
Automorphism Group ◽  
Simple Group ◽  
Cayley Graph ◽  
Finite Simple Group ◽  
Cayley Graphs ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Edge Transitive

Abstract A graph $\Gamma $ is called $(G, s)$ -arc-transitive if $G \le \text{Aut} (\Gamma )$ is transitive on the set of vertices of $\Gamma $ and the set of s-arcs of $\Gamma $ , where for an integer $s \ge 1$ an s-arc of $\Gamma $ is a sequence of $s+1$ vertices $(v_0,v_1,\ldots ,v_s)$ of $\Gamma $ such that $v_{i-1}$ and $v_i$ are adjacent for $1 \le i \le s$ and $v_{i-1}\ne v_{i+1}$ for $1 \le i \le s-1$ . A graph $\Gamma $ is called 2-transitive if it is $(\text{Aut} (\Gamma ), 2)$ -arc-transitive but not $(\text{Aut} (\Gamma ), 3)$ -arc-transitive. A Cayley graph $\Gamma $ of a group G is called normal if G is normal in $\text{Aut} (\Gamma )$ and nonnormal otherwise. Fang et al. [‘On edge transitive Cayley graphs of valency four’, European J. Combin.25 (2004), 1103–1116] proved that if $\Gamma $ is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either $\Gamma $ is normal or G is one of the groups $\text{PSL}_2(11)$ , $\text{M} _{11}$ , $\text{M} _{23}$ and $A_{11}$ . However, it was unknown whether $\Gamma $ is normal when G is one of these four groups. We answer this question by proving that among these four groups only $\text{M} _{11}$ produces connected tetravalent 2-transitive nonnormal Cayley graphs. We prove further that there are exactly two such graphs which are nonisomorphic and both are determined in the paper. As a consequence, the automorphism group of any connected tetravalent 2-transitive Cayley graph of any finite simple group is determined.

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 On the laws of the variety se

1972 ◽  
Vol 14 (3) ◽  
pp. 364-367 ◽  
Author(s):  
Roger M. Bryant
Keyword(s):  
Positive Integer ◽  
Simple Group ◽  
Finite Simple Group ◽  
Basis Property ◽  
Finite Basis ◽  
Simple Groups ◽  
Locally Finite ◽  
Prove Theorem ◽  
Finite Basis Property ◽  
Locally Finite Variety

A group is called an s-group if it is locally finite and all its Sylow subgroups are abelian. Kovács [4] has shown that, for any positive integer e, the class se of all s-groups of exponent dividing e is a (locally finite) variety. The proof of this relies on the fact that, for any e, there are only finitely many (isomorphism classes of) non-abelian finite simple groups in se; and this is a consequence of deep results of Walter and others (see [6]). In [2], Christensen raised the finite basis question for the laws of the varieties se. It is easy to establish the finite basis property for an se which contains no non-abelian finite simple group; and Christensen gave a finite basis for the laws of the variety s30, whose only non-abelian finite simple group is PSL(2,5). Here we prove Theorem For any positive integer e, the varietysehas a finite basis for its laws.

scholarly journals Steinberg-like characters for finite simple groups

2020 ◽  
Vol 23 (1) ◽  
pp. 25-78
Author(s):  
Gunter Malle ◽  
Alexandre Zalesski
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Simple Group ◽  
Regular Representation ◽  
Simple Groups ◽  
Closed Field ◽  
Finite Simple Groups ◽  
Algebraically Closed Field ◽  
Characteristic 2 ◽  
A Finite Group

AbstractLet G be a finite group and, for a prime p, let S be a Sylow p-subgroup of G. A character χ of G is called {\mathrm{Syl}_{p}}-regular if the restriction of χ to S is the character of the regular representation of S. If, in addition, χ vanishes at all elements of order divisible by p, χ is said to be Steinberg-like. For every finite simple group G, we determine all primes p for which G admits a Steinberg-like character, except for alternating groups in characteristic 2. Moreover, we determine all primes for which G has a projective FG-module of dimension {\lvert S\rvert}, where F is an algebraically closed field of characteristic p.

scholarly journals ON INVOLUTIONS AND INDICATORS OF FINITE ORTHOGONAL GROUPS

2018 ◽  
Vol 105 (3) ◽  
pp. 380-416 ◽  
Author(s):  
GREGORY K. TAYLOR ◽  
C. RYAN VINROOT
Keyword(s):  
Asymptotic Behavior ◽  
Simple Group ◽  
Generating Functions ◽  
Finite Simple Group ◽  
Point Of View ◽  
Character Degree ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Classical Groups ◽  
Orthogonal Groups

We study the numbers of involutions and their relation to Frobenius–Schur indicators in the groups $\text{SO}^{\pm }(n,q)$ and $\unicode[STIX]{x1D6FA}^{\pm }(n,q)$. Our point of view for this study comes from two motivations. The first is the conjecture that a finite simple group $G$ is strongly real (all elements are conjugate to their inverses by an involution) if and only if it is totally orthogonal (all Frobenius–Schur indicators are 1), and we observe this holds for all finite simple groups $G$ other than the groups $\unicode[STIX]{x1D6FA}^{\pm }(4m,q)$ with $q$ even. We prove computationally that for small $m$ this statement indeed holds for these groups by equating their character degree sums with the number of involutions. We also prove a result on a certain twisted indicator for the groups $\text{SO}^{\pm }(4m+2,q)$ with $q$ odd. Our second motivation is to continue the work of Fulman, Guralnick, and Stanton on generating functions and asymptotics for involutions in classical groups. We extend their work by finding generating functions for the numbers of involutions in $\text{SO}^{\pm }(n,q)$ and $\unicode[STIX]{x1D6FA}^{\pm }(n,q)$ for all $q$, and we use these to compute the asymptotic behavior for the number of involutions in these groups when $q$ is fixed and $n$ grows.

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