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be an odd prime. We construct a
of nilpotency class two, rank seven and exponent
, such that
on the Frattini quotient
. The constructed group
is the smallest
-group with these properties, having order
, and when
our construction gives two nonisomorphic
-groups. To show that
satisfies the specified properties, we study the action of
on the octonion algebra over
, for each power
, and explore the reducibility of the exterior square of each irreducible seven-dimensional
This paper explores the possible use of Schubert cells and Schubert varieties in finite geometry, particularly in regard to the question of whether these objects might be a source of understanding of ovoids or provide new examples. The main result provides a characterization of those Schubert cells for finite Chevalley groups which have the first property (thinness) of ovoids. More importantly, perhaps this short paper can help to bridge the modern language barrier between finite geometry and representation theory. For this purpose, this paper includes very brief surveys of the powerful lattice theory point of view from finite geometry and the powerful method of indexing points of flag varieties by Chevalley generators from representation theory.
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
preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on
, 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
cannot have holomorph compound O’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.
In Bachmann [Aufbau der Geometrie aus dem Spiegelungsbegriff, Die Grundlehren der mathematischen Wissenschaften, Bd. XCVI (Springer, Berlin–Göttingen–Heidelberg, 1959)], it was shown that a finite metric plane is a Desarguesian affine plane of odd order equipped with a perpendicularity relation on lines and that the converse is also true. Sherk [‘Finite incidence structures with orthogonality’, Canad. J. Math.19 (1967), 1078–1083] generalised this result to characterise the finite affine planes of odd order by removing the ‘three reflections axioms’ from a metric plane. We show that one can obtain a larger class of natural finite geometries, the so-called Bruck nets of even degree, by weakening Sherk’s axioms to allow noncollinear points.
Reports of bloodstream infections caused by methicillin-resistant Staphylococcus aureus among chronic hemodialysis patients to 2 Centers for Disease Control and Prevention surveillance systems (National Healthcare Safety Network Dialysis Event and Emerging Infections Program) were compared to evaluate completeness of reporting. Many methicillin-resistant S. aureus bloodstream infections identified in hospitals were not reported to National Healthcare Safety Network Dialysis Event.
Infect. Control Hosp. Epidemiol. 2016;37(2):205–207
Little information exists regarding how accurately emergency physicians (EPs) predict the probability of acute coronary syndrome (ACS). Our objective was to determine if EPs can accurately predict ACS in a prospectively identified cohort of emergency department (ED) patients who met enrolment criteria for a study of coronary computed tomographic angiography (CCTA) and were admitted for a “rule out ACS” protocol.
A prospective observational pilot study in an academic medical centre was carried out. EPs caring for patients with chest pain provided whole-number estimates of the probability of ACS after clinical review. This substudy was part of the now published Rule Out Myocardial Infarction/Ischemia Using Computer Assisted Tomography (ROMICAT) study, a study of CCTA and admission of patients for a rule out ACS protocol after a nondiagnostic evaluation. Predictions were grouped into probability groups based on the validated Goldman criteria. ACS was determined by an adjudication committee using American Heart Association/American College of Cardiology/European Society of Cardiology guidelines.
A total of 334 predictions were obtained for a study population with a mean age of 54 (SD 12) years, 63% of whom were male. There were 35 ACS events. EPs predicted ACS better than by chance, and increasingly higher estimates were associated with a higher incidence of ACS (p = 0.0004). The percentage of patients with ACS was 0%, 6%, 7%, and 17%, respectively, for very low, low, intermediate, and high probability groups. EPs' estimates had a sensitivity of 63% using a > 20% probability of ACS to define a positive test. Lowering this threshold to > 7% to define a test as positive increased the sensitivity of physician estimates to 89% but lowered specificity from 65% to 24%
Our data suggest that for a selected ED cohort meeting eligibility criteria for a study of CCTA, EPs predict ACS better than by chance, with an increasing proportion of patients proving to have ACS with increasing probability estimates. Lowering the estimate threshold does not result in an overall sensitivity level that is sufficient to send patients home from the ED and is associated with a poor specificity.
Finite innately transitive permutation groups include all finite quasiprimitive and primitive permutation groups. In this paper, results in the theory of quasiprimitive and primitive groups are generalised to innately transitive groups, and in particular, we extend results of Praeger and Shalev. Thus we show that innately transitive groups possess parameter bounds which are similar to those for primitive groups. We also classify the innately transitive types of quotient actions of innately transitive groups.
A finite permutation group is said to be innately transitive if it contains a transitive minimal normal subgroup. In this paper, we give a characterisation and structure theorem for the finite innately transitive groups, as well as describing those innately transitive groups which preserve a product decomposition. The class of innately transitive groups contains all primitive and quasiprimitive groups.
Let T be a subgroup of PSL (2, ℚ) generated by a pair of rational parabolic matrices P1, P2, and let ℐ be the Jøgensen number. We prove that T has a non-trivial element of finite order if and only if ℐ = 4/n2 or ℐ = 9/n2 for some non-zero integer n.
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