scholarly journals SIMPLE GROUPS, PRODUCT ACTIONS, AND GENERALIZED QUADRANGLES

2017 ◽  
Vol 234 ◽  
pp. 87-126
Author(s):  
JOHN BAMBERG ◽  
TOMASZ POPIEL ◽  
CHERYL E. PRAEGER
Keyword(s):  
Cartesian Product ◽  
Finite Geometry ◽  
Simple Groups ◽  
Generalized Quadrangles ◽  
Product Decomposition ◽  
Number Of Factors ◽  
Point Line ◽  
Set Of Points ◽  
Generic Assumption

The classification of flag-transitive generalized quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalized quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seek a classification of the point-primitive examples. Working toward this aim, we are led to investigate generalized quadrangles that admit a collineation group$G$preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on$G$, the number of factors of such a Cartesian product can be at most four. This result is then used to treat various types of primitive and quasiprimitive point actions. In particular, it is shown that$G$cannot haveholomorph compoundO’Nan–Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classes in nonabelian finite simple groups and fixities of primitive permutation groups.

scholarly journals On the Bateman–Horn conjecture for polynomials over large finite fields

2016 ◽  
Vol 152 (12) ◽  
pp. 2525-2544 ◽  
Author(s):  
Alexei Entin
Keyword(s):  
Finite Field ◽  
Affine Plane ◽  
Plane Curve ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Total Degree ◽  
Number Of Factors ◽  
Ring Of Polynomials ◽  
Image Position

We prove an analogue of the classical Bateman–Horn conjecture on prime values of polynomials for the ring of polynomials over a large finite field. Namely, given non-associate, irreducible, separable and monic (in the variable$x$) polynomials$F_{1},\ldots ,F_{m}\in \mathbf{F}_{q}[t][x]$, we show that the number of$f\in \mathbf{F}_{q}[t]$of degree$n\geqslant \max (3,\deg _{t}F_{1},\ldots ,\deg _{t}F_{m})$such that all$F_{i}(t,f)\in \mathbf{F}_{q}[t],1\leqslant i\leqslant m$, are irreducible is$$\begin{eqnarray}\displaystyle \biggl(\mathop{\prod }_{i=1}^{m}\frac{\unicode[STIX]{x1D707}_{i}}{N_{i}}\biggr)q^{n+1}(1+O_{m,\,\max \deg F_{i},\,n}(q^{-1/2})), & & \displaystyle \nonumber\end{eqnarray}$$where$N_{i}=n\deg _{x}F_{i}$is the generic degree of$F_{i}(t,f)$for$\deg f=n$and$\unicode[STIX]{x1D707}_{i}$is the number of factors into which$F_{i}$splits over$\overline{\mathbf{F}}_{q}$. Our proof relies on the classification of finite simple groups. We will also prove the same result for non-associate, irreducible and separable (over$\mathbf{F}_{q}(t)$) polynomials$F_{1},\ldots ,F_{m}$not necessarily monic in$x$under the assumptions that$n$is greater than the number of geometric points of multiplicity greater than two on the (possibly reducible) affine plane curve$C$defined by the equation$$\begin{eqnarray}\displaystyle \mathop{\prod }_{i=1}^{m}F_{i}(t,x)=0 & & \displaystyle \nonumber\end{eqnarray}$$(this number is always bounded above by$(\sum _{i=1}^{m}\deg F_{i})^{2}/2$, where$\deg$denotes the total degree in$t,x$) and$$\begin{eqnarray}\displaystyle p=\text{char}\,\mathbf{F}_{q}>\max _{1\leqslant i\leqslant m}N_{i}, & & \displaystyle \nonumber\end{eqnarray}$$where$N_{i}$is the generic degree of$F_{i}(t,f)$for$\deg f=n$.

scholarly journals Finite simple exceptional groups of Lie type in which all subgroups of odd index are pronormal

2020 ◽  
Vol 23 (6) ◽  
pp. 999-1016
Author(s):  
Anatoly S. Kondrat’ev ◽  
Natalia V. Maslova ◽  
Danila O. Revin
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups ◽  
Exceptional Groups ◽  
Odd Index ◽  
Groups Of Lie Type

AbstractA subgroup H of a group G is said to be pronormal in G if H and {H^{g}} are conjugate in {\langle H,H^{g}\rangle} for every {g\in G}. In this paper, we determine the finite simple groups of type {E_{6}(q)} and {{}^{2}E_{6}(q)} in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal.

The clinical utility of the glycemic index and its application to mixed meals

1991 ◽  
Vol 69 (1) ◽  
pp. 100-107 ◽  
Author(s):  
Clarie B. Hollenbeck ◽  
Ann M. Coulston
Keyword(s):  
Glycemic Index ◽  
Clinical Utility ◽  
Postprandial Glucose ◽  
Dependent Diabetes Mellitus ◽  
Glycemic Response ◽  
Glucose Response ◽  
Number Of Factors ◽  
Carbohydrate Intolerance ◽  
Low Glycemic Index

A classification of carbohydrate-containing foods based on their glycemic response to 50-g carbohydrate portions has recently been developed. The relative glycemic potency of many of these carbohydrate-containing foods have been compared, and these data have been published in the form of a glycemic index. It has been suggested that meals containing low glycemic index foods will result in a lower postprandial glucose response than meals with a higher glycemic index. However, whether or not these data will lead to a clinically useful reduction in postprandial hyperglycemia in individuals with carbohydrate intolerance remains controversial. In this review, we will try to delineate why we believe that the glycemic index, as currently developed, may be a specious issue. In addition, we will briefly discuss a number of factors that may explain the apparent discrepancy in viewpoints on this issue.Key words: glycemic index, noninsulin-dependent diabetes mellitus, glycemic response, dietary carbohydrate.

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 On a Theorem of P. Fong

2003 ◽  
Vol 171 ◽  
pp. 197-206
Author(s):  
Inna Korchagina
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups

AbstractThis paper is a contribution to the “revision” project of Gorenstein, Lyons and Solomon, whose goal is to produce a unified proof of the Classification of Finite Simple Groups.

The Classification of the Finite Simple Groups, Number 6

2004 ◽  
Author(s):  
Daniel Gorenstein ◽  
Richard Lyons ◽  
Ronald Solomon
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups

The Classification of Finite Simple Groups

2011 ◽  
Author(s):  
Michael Aschbacher ◽  
Richard Lyons ◽  
Stephen Smith ◽  
Ronald Solomon
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups

The Classification of the Finite Simple Groups, Number 7

2018 ◽  
Author(s):  
Daniel Gorenstein ◽  
Richard Lyons ◽  
Ronald Solomon
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups

scholarly journals Isospectral drums and simple groups

2018 ◽  
Vol 15 (04) ◽  
pp. 1850060
Author(s):  
Koen Thas
Keyword(s):  
Physical Properties ◽  
The Other ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Planar Domains ◽  
Other Hand ◽  
Isospectral Manifolds ◽  
Transplantation Technique ◽  
Rule Out

Nearly every known pair of isospectral but nonisometric manifolds — with as most famous members isospectral bounded [Formula: see text]-planar domains which makes one “not hear the shape of a drum” [M. Kac, Can one hear the shape of a drum? Amer. Math. Monthly 73(4 part 2) (1966) 1–23] — arise from the (group theoretical) Gassmann–Sunada method. Moreover, all the known [Formula: see text]-planar examples (so counter examples to Kac’s question) are constructed through a famous specialization of this method, called transplantation. We first describe a number of very general classes of length equivalent manifolds, with as particular cases isospectral manifolds, in each of the constructions starting from a given example that arises itself from the Gassmann–Sunada method. The constructions include the examples arising from the transplantation technique (and thus in particular the known planar examples). To that end, we introduce four properties — called FF, MAX, PAIR and INV — inspired by natural physical properties (which rule out trivial constructions), that are satisfied for each of the known planar examples. Vice versa, we show that length equivalent manifolds with FF, MAX, PAIR and INV which arise from the Gassmann–Sunada method, must fall under one of our prior constructions, thus describing a precise classification of these objects. Due to the nature of our constructions and properties, a deep connection with finite simple groups occurs which seems, perhaps, rather surprising in the context of this paper. On the other hand, our properties define in some sense physically irreducible pairs of length equivalent manifolds — “atoms” of general pairs of length equivalent manifolds, in that such a general pair of manifolds is patched up out of irreducible pairs — and that is precisely what simple groups are for general groups.

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