Fields of values of Fong characters

If G is a p-solvable finite group with a p-complement H and φ ∈ IBr(G), then P. Fong showed that there exists α ∈ Irr(H) such that α^G=Φ_φ. In this note we prove that α can be chosen such that the field of values index divides φ (1)p. Received: 6 May 2005     

Invariant characters and coprime actions on finite nilpotent groups

Abstract. Let S be a subgroup of SLn(R), where R is a commutative ring with identity and n \geqq 3n \geqq 3. The order of S, o(S), is the R-ideal generated by x_ij, x_ii - x_jj (i 1 j)x_{ij},\ x_{ii} - x_{jj}\ (i \neq j), where (x_ij) ? S(x_{ij}) \in S. Let En(R) be the subgroup of SLn(R) generated by the elementary matrices. The level of S, l(S), is the largest R-ideal \frak q\frak {q} with the property that S contains all the \frak q\frak {q}-elementary matrices and all conjugates of these by elements of En(R). It is clear that l(S) \leqq o(S)l(S) \leqq o(S). Vaserstein has proved that, for all R and for all n \geqq 3n \geqq 3, the subgroup S is normalized by En(R) if and only if l(S) = o(S)     

Electrochemistry of Mono- through Hexakis-adducts of C60

The first systematic electrochemical study by cyclic voltammetry (CV) and rotating-disk electrode (RDE) of the changes in redox properties of covalent fullerene derivatives ( 2 – 11 ) as a function of increasing number of addends is reported. Dialkynylmethanofullerenes 2 – 4 undergo multiple, fullerene-centered reduction steps at slightly more negative potentials than C₆₀ ( 1 ; see Table and Fig. 1). The two C-spheres in the dumbbell-shaped dimeric fullerene derivative 4 show independent, identical redox characteristics. This highlights the insulating character of the sp³-C-atoms in methanofullerenes which prevent through-bond communication of substituent effects from the methano bridge to the fullerene sphere. In the series of mono- through hexakis-adducts 5 – 11 , formed by tether-directed remote functionalization, reductions become increasingly difficult and more irreversible with increasing number of addends (see Table and Fig. 2). Whereas, in 0.1M Bu₄NPF₆/CH₂Cl₂, the first reduction of mono-adduct 5 occurs reversibly at ?1.06 V vs. the ferrocene/ferricinium couple (Fc/Fc⁺), hexakis-adduct 11 is reduced irreversibly only at ? 1.87 V. Hence, with incremental functionalization of the fullerene, the LUMO of the remaining conjugated framework is raised in energy. Reduction potentials are also dependent on the relative spatial disposition of the addends on the surface of the fullerene sphere. Observed UV/VIS spectral changes and changes in the chemical reactivity along the series 5 – 11 are in accord with the results of electrochemical measurements. Further, with increasing number of addends, the oxidation of derivatives 5 – 11 becomes more reversible. Whereas oxidations are increasingly facilitated upon going from mono-adduct 5 (+1.22 V) to tris-adduct 7 (+0.90 V), they occur at nearly the same potential (+0.95 to +0.99 V) in the higher adducts 8 – 11 . This indicates that the oxidations occur in these compounds at a common sub-structural element, for which a ‘cubic’ cyclophane is proposed (see Fig. 3). This sub-structure is fully developed in hexakis-adduct 11 .     

Structure and soft mode behavior of two chalcogenide crystals

The crystal structures of T?₃PSe₄ and T?₃AsS₄ have been determined by x-ray diffraction techniques. These isomorphous materials are layered, with orthorhombic Pcmn symmetry. Ultrasonic measurements have shown that the $\text{a}$$\text{b}$ shear mode velocity is anomalously low for both materials. Measurements at high pressure on T?₃PSe₄ have shown that the anomalous shear velocity softens to zero at 1.40 ± 0.01 GPa, indicating a second- order structural transition at that pressure.     

Derived subgroups and centers of capable groups

A group is said to be capable if it is isomorphic to the central factor group for some group . It is shown in this paper that if is finite and capable, then the index of the center in is bounded above by some function of the order of the derived subgroup . If is cyclic and its elements of order are central, then, in fact, .

     

Derived Subgroups of Products of an Abelian and a Cyclic Subgroup

Let G be a finite group and suppose that G = AB, where A andB are abelian subgroups. By a theorem of Ito, the derived subgroupG' is known to be abelian. If either of the subgroups A or Bis cyclic, then more can be said. The paper shows, for example,that G'/(G'A) is isomorphic to a subgroup of B in this case.     

Character degree sets that do not bound the class of a -group

Suppose that we are given a set of powers of a prime and that . A technique is presented that enables the construction of a -group of specified nilpotence class such that its set of irreducible character degrees is exactly . If , then this can be done for and if , then the only requirement is .