Finite groups of low essential dimension

Variants of some of the Brauer-Fowler theorems

Let G be a finite 2-generated non-cyclic group. The spread of G is the largest integer k such that for any nontrivial elements x1,…,xk, there exists y G ∈ G such that G = 〈xi,y〉 for all i. The more restrictive notion of uniform spread, denoted u(G), requires y to be chosen from a fixed conjugacy class of G, and a theorem of Breuer, Guralnick and Kantor states that u(G) ⩾ 2 for every non-abelian finite simple group G. For any group with u(G) ⩾ 1, we define the uniform domination number γu(G) of G to be the minimal size of a subset S of conjugate elements such that for each nontrivial x ∈ G there exists y ∈ S with G = 〈x, y〉 (in this situation, we say that S is a uniform dominating set for G). We introduced the latter notion in a recent paper, where we used probabilistic methods to determine close to best possible bounds on γu(G) for all simple groups G.In this paper we establish several new results on the spread, uniform spread and uniform domination number of finite groups and finite simple groups. For example, we make substantial progress towards a classification of the simple groups G with γu (G) = 2, and we study the associated probability that two randomly chosen conjugate elements form a uniform dominating set for G. We also establish new results concerning the 2-generation of soluble and symmetric groups, and we present several open problems.

Originating from a summer school taught by the authors, this concise treatment includes many of the main results in the area. An introductory chapter describes the fundamental results on linear algebraic groups, culminating in the classification of semisimple groups. The second chapter introduces more specialized topics in the subgroup structure of semisimple groups and describes the classification of the maximal subgroups of the simple algebraic groups. The authors then systematically develop the subgroup structure of finite groups of Lie type as a consequence of the structural results on algebraic groups. This approach will help students to understand the relationship between these two classes of groups. The book covers many topics that are central to the subject, but missing from existing textbooks. The authors provide numerous instructive exercises and examples for those who are learning the subject as well as more advanced topics for research students working in related areas.

The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q,G) using the descending central series of free groups. If G is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this thesis we introduce natural subspaces B(q,G)_p of B(q,G) defined for a fixed prime p. We show that B(q,G) is stably homotopy equivalent to a wedge sum of B(q,G)_p as p runs over the primes dividing the order of G. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial p-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial p-groups of rank at least 4 the space B(2,G) does not have the homotopy type of a K(π,1) space. Furthermore, we give a group theoretic condition, applicable to symmetric groups and general linear groups, which implies the space B(2,G) not having the homotopy type of a K(π,1) space. For a finite group G, we compute the complex K-theory of B(2,G) modulo torsion.

Weights for symmetric and general linear groups

Let [Formula: see text] be the prime graph associated with a finite group [Formula: see text] and [Formula: see text] be the degree pattern of [Formula: see text]. A finite group [Formula: see text] is said to be [Formula: see text]-fold [Formula: see text]-characterizable if there exist exactly [Formula: see text] nonisomorphic groups [Formula: see text] such that [Formula: see text] and [Formula: see text]. The purpose of this paper is two-fold. First, it shows that the symmetric group [Formula: see text] is [Formula: see text]-fold [Formula: see text]-charaterizable. Second, it shows that there exist many infinite families of alternating and symmetric groups, [Formula: see text] and [Formula: see text], which are [Formula: see text]-fold [Formula: see text]-characterizable with [Formula: see text].

The current literature shows no consensus regarding the difference between the dominant leg (D-Leg) and the non-dominant leg (ND-Leg) in terms of postural control. This lack of consensus could stem from motor experience (i.e., symmetric or asymmetric motricity) and/or the physiological state induced by physical exercise. This study aimed to investigate the acute effects of fatiguing exercise on postural control when standing on the D-Leg and the ND-Leg, in athletes practicing symmetric (SYM) and asymmetric (ASYM) sports. Thirty healthy male participants were recruited and divided into two groups, (SYM n = 15) and (ASYM n = 15, on the basis of the motricity induced by the sport they practice. Monopedal postural control was assessed for the D-Leg and the ND-Leg before and after the fatigue period (which consisted of repeating squats until exhaustion). A force platform was used to calculate the spatio-temporal characteristics of the displacements of the center of foot pressure (COP). A significant fatigue effect was observed in both groups on the D-Leg and the ND-Leg for all the COP parameters. There was a tendency (p = 0.06) between the ASYM and SYM groups on the D-Leg, concerning the relative increase in the COP velocity in the frontal plane after the fatigue period. The fatigue condition disturbed postural control in both the SYM and ASYM groups on the D-Leg and ND-Leg. This disturbing effect related to fatigue tends to be more marked in athletes practicing asymmetric sports than in athletes practicing symmetric sports on the D-Leg.

Let F be a field of characteristic p>0 and G be a smooth finite algebraic group over F. We compute the essential dimension edF(G;p) of G at p. That is, we show thatedF(G;p)={1,ifpdivides|G|,and0,otherwise.

Let K be any field and G be a finite group. Let G act on the rational function field by K-automorphisms defined by for any . Denote by the fixed field . Noether's problem asks whether is rational (i.e. purely transcendental) over K. We will prove that, if K is any field, p an odd prime number, and G is a nonabelian group of exponent p with or p 4 satisfying , then is rational over K. Moreover, it will be shown that is retract rational if G belongs to a much larger class of p-groups. In particular, generic G-polynomials of G-Galois extensions exist for these groups.

Noether's problem for some semidirect products

Ramification in the Inverse Galois Problem

Consider the algebraic function $\Phi _{g,n}$ that assigns to a general $g$-dimensional abelian variety an $n$-torsion point. A question first posed by Klein asks: What is the minimal $d$ such that, after a rational change of variables, the function $\Phi _{g,n}$ can be written as an algebraic function of $d$ variables? Using techniques from the deformation theory of $p$-divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $p$-dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $p$-dimension of congruence covers of the moduli space of genus $g$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety $M$ is proper. As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety.

Essential dimension of group schemes over a local scheme

Quantum Drinfeld Hecke algebras extend both Lusztig's graded Hecke algebras and the symplectic reflection algebras of Etingof and Ginzburg to the quantum setting. A quantum (or skew) polynomial ring is generated by variables which commute only up to a set of quantum parameters. Certain finite groups may act by graded automorphisms on a quantum polynomial ring and quantum Drinfeld Hecke algebras deform the natural semi-direct product. We classify these algebras for the infinite family of complex reflection groups acting in arbitrary dimension. We also classify quantum Drinfeld Hecke algebras in arbitrary dimension for the infinite family of mystic reflection groups of Kirkman, Kuzmanovich, and Zhang, who showed they satisfy a Shephard-Todd-Chevalley theorem in the quantum setting. Using a classification of automorphisms of quantum polynomial rings in low dimension, we develop tools for studying quantum Drinfeld Hecke algebras in 3 dimensions. We describe the parameter space of such algebras using special properties of the quantum determinant in low dimension; although the quantum determinant is not a homomorphism in general, it is a homomorphism on the finite linear groups acting in dimension 3.

From time immemorial, the search for healing has been an essential and universal dimension of human life. Human beings are motivated by the natural and spontaneous instinct to preserve life in its entirety, especially when health is threatened by, sickness or disease. The mission to heal belongs to all members of the human family, regardless of gender, race, age, or religion. The Church bases her mission to heal on this prerogative and on Jesus' mandate to his disciples to preach and heal. This study is a theological and pastoral analysis of the Church's involvement in the healing ministry among the Banyankore of the Archdiocese of Mbarara. The research investigates the Church's pastoral activity in this regard, examining the successes but also the challenges encountered. From the hypothesis that a theological and pastoral analysis of the Church's approach to the practice of heating in the context of Christian living today, will help to develop an integrated healing mission for the Church in the Archdiocese of Mbarara, the study set out to investigate the Church's contribution to the healing ministry at the local or diocesan level. It is an attempt to discover how best the Church can use an integrated approach to healing to fulfill Christ's legacy. As illustrated in Chapter one, the present study uses the contextual approach to theology inspired by Stephen Bevans' Anthropological and Synthetic models of contextual theology, and Theresa Okure's Incarnational paradigm as theological key to inculturating the Church's healing mission. The local Church is perceived as agent and mediator of healing. The present study takes seriously people's cultural and native practices of healing, while at the same time acknowledging the contribution of other healing traditions. Grounded in the above approach, Chapter two of the thesis looks at the Church's past and present approach to healing in the Archdiocese of Mbarara. It becomes apparent that the Church's ministry of healing at this level lays greater emphasis on the medical model - through health care services offered hospitals, dispensaries, medical clinics. Yet this approach alone is Insufficient to care for all the sick and afflicted, and does not treat sicknesses that are not physiological in nature. People search for alternatives, thus showing that there is a need for a more integrated approach to healing. Chapter three studies the Banyankore native concepts and practices of healing. The study reveals that because of their holistic world view, many sick Banyankore are attracted to native practices of healing. This discovery further emphasizes the need for integration: some of the native beliefs and practices of healing could enlighten the Church's healing ministry. Chapter four is a christological analysis of the healing dimension in the various African faces of Christ. All the dimensions contribute toward the image of Christ the Divine Healer. In other words, Christ's healing ministry as presented in the New Testament, and the Gospels in particular, finds expression in the…