Elements in finite classical groups whose powers have large 1-Eigenspaces

We estimate the proportion of several classes of elements in finite classical groups which are readily recognised algorithmically, and for which some power has a large fixed point subspace and acts irreducibly on a complement of it. The estimates are used in complexity analyses of new recognition algorithms for finite classical groups in arbitrary characteristic

Identifying long cycles in finite alternating and symmetric groups acting on subsets

Let $H$ be a permutation group on a set $\Lambda$, which is permutationally isomorphic to a finite alternating or symmetric group $A_n$ or $S_n$ acting on the $k$-element subsets of points from $\{1,\ldots,n\}$, for some arbitrary but fixed $k$. Suppose moreover that no isomorphism with this action is known. We show that key elements of $H$ needed to construct such an isomorphism $\varphi$, such as those whose image under $\varphi$ is an $n$-cycle or $(n-1)$-cycle, can be recognised with high probability by the lengths of just four of their cycles in $\Lambda$.Comment: 45 page

Permutation groups, simple groups and sieve methods

We show that the number of integers n ≤ x which occur as indices of subgroups of nonabelian finite simple groups, excluding that of An-1 in An, is ∼ hx/log x, for some given constant h. This might be regarded as a noncommutative analogue of the Prime Number Theorem (which counts indices n ≤ x of subgroups of abelian simple groups). We conclude that for most positive integers n, the only quasiprimitive permutation groups of degree n are Sn and An in their natural action. This extends a similar result for primitive permutation groups obtained by Cameron, Neumann and Teague in 1982. Our proof combines group-theoretic and number-theoretic methods. In particular, we use the classification of finite simple groups, and we also apply sieve methods to estimate the size of some interesting sets of primes

A Simplified Analysis of Radiant Heat Loss Through Projecting Fenestration Products

© 2001 ASHRAE (www.ashrae.org). Published in ASHRAE Transactions, Volume 107, Part 1. For personal use only. Additional reproduction, distribution, or transmission in either print or digital form is not permitted without ASHRAE's prior written permissioCurrent window analysis algorithms can deal with many features, including low-e coatings and substitute fill gases. These methods were developed for products with planar glazings. Results can be generated for projecting products such as greenhouse windows, but the indoor-side heat transfer coefficient must be reduced to reflect differences in convection and radiant exchange for this geometry. Two simplified models are developed for radiant heat loss to projecting windows and are shown to agree well with a pseudo three-dimensional multi-element computer-based calculation. It is confirmed that the indoor-side heat transfer coefficient does not need to be accurately known to characterize a well-insulated window. More research is needed to quantify indoor-side convective heat loss before radiant exchange models can be verified and projecting products can be well characterized in general.Natural Sciences and Engineering Research Council of Canada || Natural Resources Canad

Linear spaces with a line-transitive point-imprimitive automorphism group and Fang-Li parameter gcd(k,r) at most eight

In 1991, Weidong Fang and Huiling Li proved that there are only finitely many non-trivial linear spaces that admit a line-transitive, point-imprimitive group action, for a given value of gcd(k,r), where k is the line size and r is the number of lines on a point. The aim of this paper is to make that result effective. We obtain a classification of all linear spaces with this property having gcd(k,r) at most 8. To achieve this we collect together existing theory, and prove additional theoretical restrictions of both a combinatorial and group theoretic nature. These are organised into a series of algorithms that, for gcd(k,r) up to a given maximum value, return a list of candidate parameter values and candidate groups. We examine in detail each of the possibilities returned by these algorithms for gcd(k,r) at most 8, and complete the classification in this case.Comment: 47 pages Version 1 had bbl file omitted. Apologie

On the frequency of permutations containing a long cycle

A general explicit upper bound is obtained for the proportion $P(n,m)$ of elements of order dividing $m$, where $n-1 \le m \le cn$ for some constant $c$, in the finite symmetric group $S_n$. This is used to find lower bounds for the conditional probabilities that an element of $S_n$ or $A_n$ contains an $r$-cycle, given that it satisfies an equation of the form $x^{rs}=1$ where $s\leq3$. For example, the conditional probability that an element $x$ is an $n$-cycle, given that $x^n=1$, is always greater than 2/7, and is greater than 1/2 if $n$ does not divide 24. Our results improve estimates of these conditional probabilities in earlier work of the authors with Beals, Leedham-Green and Seress, and have applications for analysing black-box recognition algorithms for the finite symmetric and alternating groups