On a generalization of a theorem of Burnside

summary:A theorem of Burnside asserts that a finite group $G$ is \mbox {$p$-nilpotent} if for some prime $p$ a Sylow \mbox {$p$-subgroup} of $G$ lies in the center of its normalizer. In this paper, let $G$ be a finite group and $p$ the smallest prime divisor of $|G|$, the order of $G$. Let $P\in {\rm Syl}_p(G)$. As a generalization of Burnside's theorem, it is shown that if every non-cyclic \mbox {$p$-subgroup} of $G$ is self-normalizing or normal in $G$ then $G$ is solvable. In particular, if $P\ncong \langle a,b\vert a^{p^{n-1}}=1,b^2=1, b^{-1}ab=a^{1+{p^{n-2}}}\rangle$, where $n\geq 3$ for $p>2$ and $n\geq 4$ for $p=2$, then $G$ is \mbox {$p$-nilpotent} or \mbox {$p$-closed}

On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups

Let G be a finite group and (G) the set of the numbers of conjugates of noncyclic proper subgroups of G. We prove that (1) if |(G)|≤2, then G is solvable, and (2) G is a nonsolvable group with |(G)|=3 if and only if G≅PSL(2,5) or PSL(2,13) or SL(2,5) or SL(2,13)

Phase Fluctuation Analysis in Functional Brain Networks of Scaling EEG for Driver Fatigue Detection

The characterization of complex patterns arising from electroencephalogram (EEG) is an important problem with significant applications in identifying different mental states. Based on the operational EEG of drivers, a method is proposed to characterize and distinguish different EEG patterns. The EEG measurements from seven professional taxi drivers were collected under different states. The phase characterization method was used to calculate the instantaneous phase from the EEG measurements. Then, the optimization of drivers’ EEG was realized through performing common spatial pattern analysis. The structures and scaling components of the brain networks from optimized EEG measurements are sensitive to the EEG patterns. The effectiveness of the method is demonstrated, and its applicability is articulated.</p

Finite groups with few vanishing elements

Let G be a finite group, and Irr(G) the set of irreducible complex characters of G. We say that an element g G is a vanishing element of G if there exists χ in Irr(G) such that χ(g)= 0. Let Van(G) denote the set of vanishing elements of G, that is, Van(G)= {g G|χ(g)=0 for some χ Irr (G)}. In this paper, we investigate the finite groups G with the following property: Van(G) contains at most four conjugacy classes of G

Fabrication of densified wood via synergy of chemical pretreatment, hot-pressing and post mechanical fixation

Article describes examination of the appearance, color, chemical composition, and physiology and mechanical properties of densified Abies wood before and after densification treatment using a colorimeter, FTIR and mechanical testing machine

Tetragonal Mexican-Hat Dispersion and Switchable Half-Metal State with Multiple Anisotropic Weyl Fermions in Penta-Graphene

In past decades, the ever-expanding library of 2D carbon allotropes has yielded a broad range of exotic properties for the future carbon-based electronics. However, the known allotropes are all intrinsic nonmagnetic due to the paired valence electrons configuration. Based on the reported 2D carbon structure database and first-principles calculations, herein we demonstrate that inherent ferromagnetism can be obtained in the prominent allotrope, penta-graphene, which has an unique Mexican-hat valence band edge, giving rise to van Hove singularities and electronic instability. Induced by modest hole-doping, being achievable in electrolyte gate, the semiconducting pentagraphene can transform into different ferromagnetic half-metals with room temperature stability and switchable spin directions. In particular, multiple anisotropic Weyl states, including type-I and type-II Weyl cones and hybrid quasi Weyl nodal loop, can be found in a sizable energy window of spin-down half-metal under proper strains. These findings not only identify a promising carbon allotrope to obtain the inherent magnetism for carbon-based spintronic devices, but highlight the possibility to realize different Weyl states by combining the electronic and mechanical means as well

On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups

Let be a finite group and NC( ) the set of the numbers of conjugates of noncyclic proper subgroups of . We prove that (1) if |NC( )| ≤ 2, then is solvable, an