Abstract For a solvable group, a theorem of Gaschutz shows that F ⁢ ( G ) / Φ ⁢ ( G ) {F(G)/\Phi(G)} is a direct sum of irreducible G-modules and a faithful G / F ⁢ ( G ) {G/F(G)} -module. If each of these irreducible modules is primitive, we show that every non-vanishing element of G lies in F ⁢ ( G ) {F(G)} .

We prove the following result. Let p be a prime, and let G be a finite p-solvable group that is a direct product of two non-cyclic subgroups of coprime order, and let V be some faithful irreducible module for G over some field in characteristic p. Then G has a regular orbit, that is, there exists some v∈V such that CG(v)=1.

Let a finite group [Formula: see text] act coprimely on a finite group [Formula: see text]. The Glauberman–Isaacs correspondence [Formula: see text] is a bijection from the set of [Formula: see text]-invariant irreducible characters of [Formula: see text] onto the set [Formula: see text] of irreducible characters of the centralizer of [Formula: see text] in [Formula: see text]. Let [Formula: see text] be a subgroup of [Formula: see text]. Composing from left to right, it follows that [Formula: see text] is an injection from [Formula: see text] into [Formula: see text]. We show that, in some cases, the map can be defined via the actions of some subgroups of [Formula: see text] containing [Formula: see text] on the centralizers in [Formula: see text] of some other such subgroups. We also show in many instances, such as [Formula: see text] odd or [Formula: see text] supersolvable and [Formula: see text] solvable, that this map is independent of the overgroup [Formula: see text].

We characterize those groups G and vector spaces V such that V is a faithful irreducible G-module and such that each v in V is centralized by a G-conjugate of a fixed non-identity element of the Fitting subgroup F(G) of G. We also determine those V and G for which V is a faithful quasi primitive G-module and F(G) has no regular orbit. We do use these to show in some cases that a non-vanishing element lies in F(G).

In a solvable group G, if p2 does not divide χ(1) for all χ∈Irr(G), then we prove that |G:F(G)|p≤p2. This bound is best possible.

Our eyes are in continuous motion. Even when we attempt to fix our gaze, we produce so called “fixational eye movements”, which include microsaccades, drift, and ocular microtremor (OMT). Microsaccades, the largest and fastest type of fixational eye movement, shift the retinal image from several dozen to several hundred photoreceptors and have equivalent physical characteristics to saccades, only on a smaller scale (Martinez-Conde, Otero-Millan & Macknik, 2013). OMT occurs simultaneously with drift and is the smallest of the fixational eye movements (∼1 photoreceptor width, >0.5 arcmin), with dominant frequencies ranging from 70 Hz to 103 Hz (Martinez-Conde, Macknik & Hubel, 2004). Due to OMT’s small amplitude and high frequency, the most accurate and stringent way to record it is the piezoelectric transduction method. Thus, OMT studies are far rarer than those focusing on microsaccades or drift. Here we conducted simultaneous recordings of OMT and microsaccades with a piezoelectric device and a commercial infrared video tracking system. We set out to determine whether OMT could help to restore perceptually faded targets during attempted fixation, and we also wondered whether the piezoelectric sensor could affect the characteristics of microsaccades. Our results showed that microsaccades, but not OMT, counteracted perceptual fading. We moreover found that the piezoelectric sensor affected microsaccades in a complex way, and that the oculomotor system adjusted to the stress brought on by the sensor by adjusting the magnitudes of microsaccades.

Our eyes continually move even while we fix our gaze. Fixational eye movements include microsaccades, drifts and ocular microtremor (OMT), a high frequency tremor of the eye. Microsaccades and OMT are the fixational eye movements with respectively largest and smallest amplitudes (up to several dozen photoreceptors in the case of microsaccades, and approximately one photoreceptor in the case of OMT). We simultaneously recorded the eye movements of human subjects with a piezoelectric device (adapted from the piezoelectric transducer technique developed by Bengi and Thomas (1968)) and a commercial infrared video tracking system (EyeLink II). The piezoelectric system could detect very small changes in eye velocity, such as those due to OMT. Both recording systems moreover allowed microsaccade detection, thus allowing the direct comparison of microsaccade dynamics between systems. Here we present experimental results concerning the effect of the piezoelectric sensors on the amplitude and other parameters of microsaccades during fixation. The application of a piezoelectric sensor resulted in decreased microsaccade amplitudes (as measured with EyeLink II). Normal amplitudes were restored upon sensor removal. Application of the sensor in one eye only resulted in a simultaneous increase of microsaccade amplitudes in the other eye. We also conducted a perceptual experiment to test the potential role of tremor on target visibility. Whereas microsaccades are known to counteract visual fading, it is not known whether tremor also contributes to visibility during fixation. We found that microsaccades were effective in counteracting fading in both foveal and peripheral target locations, in agreement with previous studies (Martinez-Conde et al., 2006; Troncoso et al., 2008). There was no correlation between OMT and the reappearance of a faded stimulus.

A 51-year-old woman presented with anosognosia for hemiplegia (AHP), neglect, and a complete loss of vision, for which she was almost immediately aware. Neuroimaging studies revealed intracranial hemorrhages in the medial temporal lobes bilaterally, extending back to the occipital cortex, but sparing the calcarine cortex. A large right frontal-parietal hemorrhage which extended to the posterior body of the corpus callosum was also observed. The patient's vision slowly improved, and by 11 months post onset, formal visual fields revealed improvement primarily in the left upper quadrants only. In contrast, resolution of her AHP occurred between the 26th and 31st day post onset. Awareness of motor impairment was correlated with her ability to initiate finger tapping in her left hemiplegic/paretic hand. During the time she was unaware of her motor deficits but aware of her visual impairments, her dreams did not reflect concerns over visual or motor limitations. The findings support a "modular" theory of anosognosia.

1. (i) Suppose K is a conjugacy class of Sn contained in An; then K is called split if K is a union of two conjugacy classes of An. Show that the number of split conjugacy classes contained in An is equal to the number of characters χ ∈ Irr(Sn) such that χAn is not irreducible. (Hint. Consider the vector space of class functions on An which are invariant under conjugation by the transposition (12).)

To the Editor: The comments following the recent case report1 of a patient with an uncommon presentation of a common (post chiropractic internal carotid artery [ICA] dissection) condition are appreciated. Our patient presented with ICA dissection and transient ischemia causing ipsilateral hemispheric language aphasic errors, and focal localized effects on the optic nerve and ciliary ganglion within the ipsilateral orbit. These 2 intra-orbital structures share a common blood supply from the ICA and ipsilateral ophthalmic artery. Wang et al in the post discussion comments section address three important take-home messages: the importance of noting both afferent and efferent pupil defects, localization of same, and the dangers of chiropractic neck manipulation. The first and second issues encompass the most important pearls for teaching purposes and for accurate neuro-anatomic localization. The unusual combination of concurrent afferent and efferent pupillary defects in the absence of ocular motility defects or homonymous field loss offers a unique localization suggesting ciliary ganglionopathy. The relative afferent pupillary defect (RAPD) is commonly accepted as the sine qua non of optic neuropathy or large retinal pathology. However, let us not forget, that optic tractopathy with its expected contralesional homonymous field loss may also present with contralesional RAPD due to asymmetric nasal and temporal retinal contributions to the optic tract.2 Wang et al (comments) accurately review that most cases of the PION syndrome are from systemic blood pressure drop with bilateral optic neuropathy. The comments correctly remind us of the PION syndrome, which as a clinical entity, has little bearing on this case with unilateral ICA dissection, recurrent transient monocular visual loss (TMVL) with RAPD. Although the posterior aspect of the optic nerve is involved with an ischemic process, the term PION refers to a distinct clinical entity, not necessarily appropriate in this localization. This case in contrast, demonstrates a unique localization based on the combination of pupillary abnormalities. We can all agree that the presence of the RAPD in this case suggests ipsilateral optic neuropathy with associated transient RAPD and ipsilateral visual loss. It can also be agreed that the efferent pupillary defect in isolation without any ocular motility defects, originates from the pupillary fibers running with CN3. More on point and for clarification, this case highlights that when found in combination, a recurrent transient afferent pupillary defect (RAPD) with ipsilateral transient monocular visual loss, concurrent with an isolated efferent pupil defect (internal ophthalmoplegia—mydriasis—without other CN3 motility loss), should force us to localize beyond merely that this is “within the orbit.” Generally, “orbit apex” syndromes mentioned in the same comments as a possible localization, is characterized by ophthalmoplegia combined with RAPD (concurrent optic neuropathy) because of involvement of the contents of both superior orbital fissure and optic canal. Additionally, the suggested alternative mechanism of a sympathetic irritative lesion (Wang et al) in this case to explain the mydriatic pupil is problematic without the concurrent optic neuropathy with RAPD being associated with any irritative photopsias. In fact, our patient had at least 3 pre-angioplasty episodes in hospital during pressor treatment and complete reversal of all signs and symptoms, thus leading to a presumed mechanism related to ophthalmic artery flow. The angioplasty was ultimately permanently successful. In this case study, the transient combined pupillary abnormalities following ICA dissection and hypo-perfusion ischemia to the ipsilateral ophthalmic artery were secondary to transient ischemia to the ciliary ganglion and adjacent optic nerve within the posterior orbit. This is the proposed mechanism for the transient ischemic tonic (mydriatic) pupil (internal ophthalmoplegia) causing the efferent pupillary defect, and the concurrent transient ipsilateral afferent pupillary defect involving the posterior optic nerve. A follow up paper will examine a hemodynamic perfusion model for this flow dependent ischemic ciliary ganglionopathy Thomas R. Wolf Stanley Iyadurai St. Louis, Missouri

OBJECTIVE:A patient with cervical internal carotid artery (ICA) dissection presented with visual loss and a mydriatic pupil that resolved after angioplasty and stenting.CLINICAL PRESENTATION:A 49-year-old woman presented with a unilateral dilated tonic pupil and transient monocular visual loss and subsequently developed speech disturbance. Angiography revealed a left cervical ICA dissection with significant luminal narrowing. The ophthalmic artery filled retrograde through external carotid artery branches and reconstituted the supraclinoid ICA. Computed tomographic perfusion showed significant hypoperfusion of the left hemisphere. Magnetic resonance imaging showed punctate boundary zone infarcts.INTERVENTION:The patient experienced pressure-dependent left hemispheric transient ischemic attacks and pressure-dependent ocular findings despite anticoagulation. She underwent uncomplicated left ICA angioplasty and stenting. The flow through the ophthalmic artery became anterograde. The tonic pupil returned to symmetry with the contralateral pupil, and the patient's symptoms resolved completely.CONCLUSION:Cervical ICA dissection can manifest with a tonic mydriatic pupil. Treatment with angioplasty and stenting of the dissected segment can restore flow and resolve the pupillary abnormality. A pathophysiological mechanism for the mydriasis is proposed.

If pi is a set of primes, a finite group G is called block pi-separated if for every two distinct irreducible complex characters alpha, beta is an element of Irr(G) there is a prime p is an element of pi such that alpha and beta are in different p-blocks. The group G is called principally pi-separated if the above holds whenever beta = 1(G). Bessenrodt and Zhang conjectured that if G is a solvable principally pi-separated group then G is pi-separated. We construct a family of counter-examples to this conjecture.

If π is a set of primes, a finite group G is block π-separated if for every two distinct irreducible complex characters α, β ∈ Irr(G) there exists a prime p ∈ π such that α and β lie in different Brauer p-blocks. A group G is block separated if it is separated by the set of prime divisors of |G|. Given a set π with n different primes, we construct an example of a solvable π-group G which is block separated but it is not separated by every proper subset of π.

Let G and A be finite groups with coprime orders, and suppose that A acts on G by automorphisms. Let pi(G, A):Irr(A)(G) --> Irr(C-G(A)) be the Glauberman-Isaacs correspondence. Let B less than or equal to A and chi is an element of IrrA(G). We exhibit a counterexample to the conjecture that chi pi(G, A) is an irreducible constituent of the restriction of chi pi(G, B) to C-G(A).