On some subgroups of linear groups over F2\mathbb{F}_2 generated by elements of order 33

Let $V$ be a vector space over the field of order $2$. We investigate subgroups of the linear group $GL(V)$ which are generated by a conjugacy class $D$ of elements of order $3$ such that all $d$ in $D$ have $2$-dimensional commutator space $[V,d]$

The geometry of hyperbolic lines in polar spaces

In this paper we consider partial linear spaces induced on the point set of a polar space, but with as lines the hyperbolic lines of this polar space. We give some geometric characterizations of these and related spaces. The results have applications in group theory, in the theory of Lie algebras and in graph theory

A geometric characterization of the symplectic Lie algebra

A nonzero element $x$ in a Lie algebra $\mathfrak{g}$ with Lie product $[ , ]$ is called extremal if $[x,[x,y]]$ is a multiple of $x$ for all $y$. In this paper we characterize the (finitary) symplectic Lie algebras as simple Lie algebras generated by their extremal elements satisying the condition that any two noncommuting extremal elements $x,y$ generate an $\mathfrak{sl}_2$ and any third extremal element $z$ commutes with at least one extremal element in this $\mathfrak{sl}_2$

A geometric characterization of the classical Lie algebras

A nonzero element x in a Lie algebra g over a field F with Lie product [ , ] is called a extremal element if [x, [x, g]] is contained in Fx. Long root elements in classical Lie algebras are examples of extremal elements. Arjeh Cohen et al. initiated the investigation of Lie algebras generated by extremal elements in order to provide a geometric characterization of the classical Lie algebras generated by their long root el- ements. He and Gabor Ivanyos studied the so-called extremal geometry with as points the 1-dimensional subspaces of g generated by extremal elements of g and as lines the 2-dimensional subspaces of g all whose nonzero vectors are extremal. For simple finite dimensional g this geometry turns out to be a root shadow space of a spherical building. In this paper we show that the isomorphism type of g is determined by its extremal geometry, provided the building has rank at least 3

Recovering the Lie algebra from its extremal geometry

An element $x$ of a Lie algebra $L$ over the field $F$ is extremal if $[x,[x,L]]=Fx$. Under minor assumptions, it is known that, for a simple Lie algebra $L$, the extremal geometry ${\cal{E}}(L)$ is a subspace of the projective geometry of $L$ and either has no lines or is the root shadow space of an irreducible spherical building $\Delta$. We prove that if $\Delta$ is of simply-laced type, then $L$ is a quotient of a Chevalley algebra of the same type.Comment: 24 page

The geometry of secants in embedded polar spaces

AbstractConsider a polar space S weakly embedded in a projective space P. A secant of S is the intersection of the point set of S with a line of P spanned by two non-collinear points of S. The geometry consisting of the points of S and as lines the secants is a so-called Delta space. In this paper we give a characterization of this and some related geometries

Fracture of the tibial baseplate in bicompartmental knee arthroplasty

Introduction. Bicompartmental knee arthroplasty (BKA) addresses combined medial and patellofemoral compartment osteoarthritis, which is relatively common, and has been proposed as a bridge between unicompartmental and total knee arthroplasty (TKA). Case Presentation. We present the case report of a young active man treated with BKA after unsuccessful conservative therapy. Four years later, loosening with fracture of the tibial baseplate was identified and the patient was revised to TKA. Discussion. Although our case is only the second fractured tibial baseplate to be reported, we believe that the modular titanium design, with two fixation pegs, is too thin to withstand daily cyclic loading powers. Light daily routine use, rather than high-impact sports, is therefore advised. Failures may also be related to the implant being an early generation and known to be technically complex, with too few implant sizes. We currently use TKA for the treatment of medial and patellofemoral compartment osteoarthritis

Graphs related to Held's simple group

AbstractWe analyze the permutation representations of low degree of Held's simple group He. We also determine its primitive multiplicity free permutation representations and show that there is no graph on which it or its automorphism group acts as a distance transitive group of automorphisms. In doing so, we supply a computer-free construction of He

Characterizations of symplectic polar spaces

A polar space S is said to be symplectic if it admits an embedding e in a projective geometry PG(V) such that the e-image e(S) of S is defined by an alternating form of V. In this paper we characterize symplectic polar spaces in terms of their incidence properties, with no mention of peculiar properties of their embeddings. This is relevant especially when S admits different (non isomorphic) embeddings, as it is the case (precisely) when S is defined over a field of characteristic 2.Comment: 20 pages/extensively revise