Biomimetic epoxidation of alkenes with sodium periodate catalyzed by tetraphenylporphyrinatomanganese(III) chloride supported on multiwall carbon nanotubes

The biomimetic epoxidation of alkenes catalyzed by tetraphenylporphyrinatomanganese(III) chloride, [Mn(TPP)Cl], immobilized on multiwall carbon nanotubes modified with 4-aminopyridine and 4-aminophenol is reported. These heterogenized catalysts were used as efficient and reusable catalysts for epoxidation of a variety of cyclic and linear alkenes with sodium periodate under mild conditions. The catalysts, [Mn(TPP)Cl@amine-MWCNT], were characterized by physico-chemical and spectroscopic methods. The effect of ultrasonic irradiation on these catalytic systems was also investigated. The catalysts were reused several times without loss of their activity.     

On finite groups with exactly seven element centralizers

For a finite groupG, #Cent(G) denotes the number of centralizers of its elements. A groupG is calledn-centralizer if #Cent(G) =n, and primitive n-centralizer if $\# Cent(G) = \# Cent\left( {\frac{G}{{Z(G)}}} \right) = n$ . The first author in [1], characterized the primitive 6-centralizer finite groups. In this paper we continue this problem and characterize the primitive 7-centralizer finite groups. We prove that a finite groupG is primitive 7-centralizer if and only if $\frac{G}{{Z(G)}} \cong D_{10} $ orR, whereR is the semidirect product of a cyclic group of order 5 by a cyclic group of order 4 acting faithfully. Also, we compute#Cent(G) for some finite groups, using the structure ofG modulu its center.