The length and depth of algebraic groups

Let $G$ be a connected algebraic group. An unrefinable chain of $G$ is a chain of subgroups $G = G_0 > G_1 > \cdots > G_t = 1$, where each $G_i$ is a maximal connected subgroup of $G_{i-1}$. We introduce the notion of the length (respectively, depth) of $G$, defined as the maximal (respectively, minimal) length of such a chain. Working over an algebraically closed field, we calculate the length of a connected group $G$ in terms of the dimension of its unipotent radical $R_u(G)$ and the dimension of a Borel subgroup $B$ of the reductive quotient $G/R_u(G)$. In particular, a simple algebraic group of rank $r$ has length $\dim B + r$, which gives a natural extension of a theorem of Solomon and Turull on finite quasisimple groups of Lie type. We then deduce that the length of any connected algebraic group $G$ exceeds $\frac{1}{2} \dim G$. We also study the depth of simple algebraic groups. In characteristic zero, we show that the depth of such a group is at most $6$ (this bound is sharp). In the positive characteristic setting, we calculate the exact depth of each exceptional algebraic group and we prove that the depth of a classical group (over a fixed algebraically closed field of positive characteristic) tends to infinity with the rank of the group. Finally we study the chain difference of an algebraic group, which is the difference between its length and its depth. In particular we prove that, for any connected algebraic group $G$, the dimension of $G/R(G)$ is bounded above in terms of the chain difference of $G$.Comment: 18 pages; to appear in Math.

Base sizes for simple groups and a conjecture of Cameron

Let G be a permutation group on a finite set ?. A base for G is a subset B C_ ? whose pointwise stabilizer in G is trivial; we write b(G) for the smallest size of a base for G. In this paper we prove that b(G) ? if G is an almost simple group of exceptional Lie type and is a primitive faithful G-set. An important consequence of this result, when combined with other recent work, is that b(G) ? 7 for any almost simple group G in a non-standard action, proving a conjecture of Cameron. The proof is probabilistic and uses bounds on fixed point ratios

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

Permutation groups with restricted stabilizers

Fix a positive integer $d$ and let $\Gamma_d$ be the class of finite groups without sections isomorphic to the alternating group $A_d$. The groups in $\Gamma_d$ were studied by Babai, Cameron and P\'{a}lfy in the 1980s and they determined bounds on the order of a primitive permutation group with this property, which have found a wide range of applications. Subsequently, results on the base sizes of such groups were also obtained. In this paper we replace the structural conditions on the group by restrictions on its point stabilizers, and we obtain similar, and sometimes stronger conclusions. For example, we prove that there is a linear function $f$ such that the base size of any finite primitive group with point stabilizers in $\Gamma_d$ is at most $f(d)$. This generalizes a recent result of the first author on primitive groups with solvable point stabilizers. For non-affine primitive groups we obtain stronger results, assuming only that stabilizers of $c$ points lie in $\Gamma_d$. We also show that if $G$ is any permutation group of degree $n$ whose $c$-point stabilizers lie in $\Gamma_d$, then $|G| \leqslant ((1+o_c(1))d/e)^{n-1}$. This asymptotically extends and improves a $d^{n-1}$ upper bound on $|G|$ obtained by Babai, Cameron and P\'{a}lfy assuming $G \in \Gamma_d$.Comment: 19 pages; to appear in Journal of Algebra (special issue in memory of Jan Saxl

Reflexive and preparatory selection and suppression of salient information in the right and left posterior parietal cortex

Attentional cues can trigger activity in the parietal cortex in anticipation of visual displays, and this activity may, in turn, induce changes in other areas of the visual cortex, hence, implementing attentional selection. In a recent TMS study [Mevorach, C., Humphreys, G. W., & Shalev, L. Opposite biases in salience-based selection for the left and right posterior parietal cortex. Nature Neuroscience, 9, 740-742, 2006b], it was shown that the posterior parietal cortex (PPC) can utilize the relative saliency (a nonspatial property) of a target and a distractor to bias visual selection. Furthermore, selection was lateralized so that the right PPC is engaged when salient information must be selected and the left PPC when the salient information must be ignored. However, it is not clear how the PPC implements these complementary forms of selection. Here we used on-line triple-pulse TMS over the right or left PPC prior to or after the onset of global/local displays. When delivered after the onset of the display, TMS to the right PPC disrupted the selection of the more salient aspect of the hierarchical letter. In contrast, left PPC TMS delivered prior to the onset of the stimulus disrupted responses to the lower saliency stimulus. These findings suggest that selection and suppression of saliency, rather than being "two sides of the same coin," are fundamentally different processes. Selection of saliency seems to operate reflexively, whereas suppression of saliency relies on a preparatory phase that "sets up" the system in order to effectively ignore saliency

Testing three hypotheses about effects of sensitive-insensitive parenting on telomeres.

Telomeres are the protective DNA-protein sequences appearing at the ends of chromosomes; they shorten with each cell division and are considered a biomarker of aging. Shorter telomere length and greater erosion have been associated with compromised physical and mental health and are hypothesized to be affected by early life stress. In the latter case, most work has relied on retrospective measures of early life stressors. The Dutch research (n = 193) presented herein tested 3 hypotheses prospectively regarding effects of sensitive-insensitive parenting during the first 2.5 years on telomere length at age 6, when first measured, and change over the following 4 years. It was predicted that (1) less sensitive parenting would predict shorter telomeres and greater erosion and that such effects would be most pronounced in children (2) exposed to prenatal stress and/or (3) who were highly negatively emotional as infants. Results revealed, only, that prenatal stress amplified parenting effects on telomere change-in a differential-susceptibility-related manner: Prenatally stressed children displayed more erosion when they experienced insensitive parenting and less erosion when they experienced sensitive parenting. Mechanisms that might initiate greater postnatal plasticity as a result of prenatal stress are highlighted and future work outlined. (PsycINFO Database Record (c) 2020 APA, all rights reserved)

Contextual Object Detection with a Few Relevant Neighbors

A natural way to improve the detection of objects is to consider the contextual constraints imposed by the detection of additional objects in a given scene. In this work, we exploit the spatial relations between objects in order to improve detection capacity, as well as analyze various properties of the contextual object detection problem. To precisely calculate context-based probabilities of objects, we developed a model that examines the interactions between objects in an exact probabilistic setting, in contrast to previous methods that typically utilize approximations based on pairwise interactions. Such a scheme is facilitated by the realistic assumption that the existence of an object in any given location is influenced by only few informative locations in space. Based on this assumption, we suggest a method for identifying these relevant locations and integrating them into a mostly exact calculation of probability based on their raw detector responses. This scheme is shown to improve detection results and provides unique insights about the process of contextual inference for object detection. We show that it is generally difficult to learn that a particular object reduces the probability of another, and that in cases when the context and detector strongly disagree this learning becomes virtually impossible for the purposes of improving the results of an object detector. Finally, we demonstrate improved detection results through use of our approach as applied to the PASCAL VOC and COCO datasets

Primitive Words, Free Factors and Measure Preservation

Let F_k be the free group on k generators. A word w \in F_k is called primitive if it belongs to some basis of F_k. We investigate two criteria for primitivity, and consider more generally, subgroups of F_k which are free factors. The first criterion is graph-theoretic and uses Stallings core graphs: given subgroups of finite rank H \le J \le F_k we present a simple procedure to determine whether H is a free factor of J. This yields, in particular, a procedure to determine whether a given element in F_k is primitive. Again let w \in F_k and consider the word map w:G x G x ... x G \to G (from the direct product of k copies of G to G), where G is an arbitrary finite group. We call w measure preserving if given uniform measure on G x G x ... x G, w induces uniform measure on G (for every finite G). This is the second criterion we investigate: it is not hard to see that primitivity implies measure preservation and it was conjectured that the two properties are equivalent. Our combinatorial approach to primitivity allows us to make progress on this problem and in particular prove the conjecture for k=2. It was asked whether the primitive elements of F_k form a closed set in the profinite topology of free groups. Our results provide a positive answer for F_2.Comment: This is a unified version of two manuscripts: "On Primitive words I: A New Algorithm", and "On Primitive Words II: Measure Preservation". 42 pages, 14 figures. Some parts of the paper reorganized towards publication in the Israel J. of Mat