• Korean Journal of Mathematics
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• 제5권2호
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• pp.185-193
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• 1997

In this paper, we study the $BP_*$-module structure of $BP_*$(BG) mod $(p,v_1,{\cdots})^2$ for non abelian groups of the order $p^3$. We know $grBP_*(BG)=BP_*{\otimes}H(H_*(BG);Q_1){\oplus}BP^*/(p,v_1){\otimes}ImQ_1$. The similar fact occurs for $BP_*$-homology $grBP_*(BG)=BP_*s^{-1}H(H_*(BG);Q_1){\oplus}BP_*/(p,v)s^{-1}H^{odd}(BG)$ by using the spectral sequence $E^{*,*}_2=Ext_{BP^*}(BP_*(BG),BP^*){\Rightarrow}BP^*(BG)$.

• Journal of Photoscience
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• 제4권1호
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• pp.7-10
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• 1997

The structures of 8-membered azalactone isomers produced from photocyclization of 2-(N,N-dibenzylamino)ethyl benzoylacetate were determined by the X-ray structure analysis to clarify the stereochemical behavior of a 1,8-biradical. The remarkable stereoselectivity in cyclization of the 1,8-biradical to form cis- and trans-isomers of the azalactone was not observed. The ring conformations were boat-chair like and dihedral angles between C$_5$- and C$_6$- phenyl groups were ca 45$\circ$ in the both isomers. The 1,8-biradicals in the transition state for the cyclization would have nearly same boat-chair like conformation and twisted configuration with the dihedral angle of ca 45$\circ$ as the corresponding isomer, and this is responsible for luck of stereoselectivity in long-chain biradical cyclization.

• 대한수학회지
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• 제61권1호
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• pp.109-132
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• 2024

The Clifford algebra of a direct sum of real quadratic spaces appears as the superalgebra tensor product of the Clifford algebras of the summands. The purpose of the current paper is to present a purely settheoretical version of the superalgebra tensor product which will be applicable equally to groups or to their non-associative analogues - quasigroups and loops. Our work is part of a project to make supersymmetry an effective tool for the study of combinatorial structures. Starting from group and quasigroup structures on four-element supersets, our superproduct unifies the construction of the eight-element quaternion and dihedral groups, further leading to a loop structure which hybridizes the two groups. All three of these loops share the same character table.

• 대한수학회보
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• 제34권1호
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• pp.1-8
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• 1997

Let G be a finiteg oroup. We denote BG a classifying space of G, which a contractible universal principal G bundle EG. The stable type of BG does not determine G up to isomorphism. A simple example [due to N. Minami]is given by $Q_{4p} \times Z/2$ and $D_{2p} \times Z/4$ where ps is an odd prime, $Q_{4p} is the generalized quarternion group of order 4p and $D_{2p}$ is the dihedral group of order 2p. However the paper [6] gives us a necessary and sufficient condition for $BG_1$ and $BG_2$ to be stably equivalent localized et pp. The local stable type of BG depends on the conjegacy classes of homomorphisms from the p-groups Q into G. This classification theorem simplifies if G has a normal sylow p-subgroup. Then the stable homotopy type depends on the Weyl group of the sylow p-subgroup.

• 충청수학회지
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• 제31권4호
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• pp.475-486
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• 2018

We study free actions of finite nonabelian groups on 3-dimensional nilmanifolds with the first homology ${\mathbb{Z}}^2{\bigoplus}{\mathbb{Z}}_2$ which yield an orbit manifold reversing fiber orientation, up to topological conjugacy. We show that those nonabelian groups are $D_4$(the dihedral group), $Q_8$(the quaternion group), and $C_8.C_4$(the $1^{st}$ non-split extension by $C_8$ of $C_4$ acting via $C_4/C_2=C_2$).

• Bulletin of the Korean Chemical Society
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• 제17권8호
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• pp.693-695
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• 1996

The steric effect on the first order hyperpolarizability of the β-nitrostyrene derivatives has been investigated by comparing the β values and the dihedral angles between the aryl and nitroalkenyl groups. In general the β value increased with the electron-donating ability of the substituent. The larger β value for 3,4-dimethoxy-β-nitrostyrene than that for p-methoxy-β-nitrostyrene has been attributed to the lower charge transfer energy for the former. The most striking substituent effect was observed in the β-methyl-β-nitrostyrene derivatives. Thus the β values for 3,4-dimethoxy- and p-methoxy derivatives of the latter decreased to near zero, probably because of the large distortion from planarity caused by the steric repulsion between the β-methyl and the aryl groups. The larger β value for p-dimethylamino-β-methyl-β-nitrostyrene has been interpreted with an increased electron-donating ability of the substituent and increased co-planarity.

• Kyungpook Mathematical Journal
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• 제63권2호
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• pp.167-173
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• 2023

Let G be a finite group and let ν(G) be the probability that two randomly selected elements of G produce a nilpotent group. In this article we show that for every positive integer n > 0, there is a finite group G such that ${\nu}(G)={\frac{1}{n}}$. We also classify all groups G with ${\nu}(G)={\frac{1}{2}}$. Further, we prove that if G is a solvable nonnilpotent group of even order, then ${\nu}(G){\leq}{\frac{p+3}{4p}}$, where p is the smallest odd prime divisor of |G|, and that equality exists if and only if $\frac{G}{Z_{\infty}(G)}$ is isomorphic to the dihedral group of order 2p where Z_∞(G) is the hypercenter of G. Finally we find an upper bound for ν(G) in terms of |G| where G ranges over all groups of odd square-free order.

• Bulletin of the Korean Chemical Society
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• 제27권9호
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• pp.1297-1304
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• 2006

A systematic study of the structure, energetics and spectral characteristics of substituted aminoketenes $R(NH_2)$C=C=O (R = H, $CH_3$, $NH_2$, OH, $OCH_3$, CH=$CH_2$, C$\equiv$CH, CN, CHO, NO, $NO_2$) which are highly reactive and transient intermediates in synthesis has been conducted by ab initio calculations at the MP2/6- 31G*//MP2/6-31G* level. Twenty four stable isomers of the eleven substituted aminoketenes having dihedral angles $\phi NH_2\sim120{^{\circ}}$ and $60^{\circ}$ have been identified and their optimized geometries and energies obtained. Electrostatic and steric effects on the molecular geometries have been analyzed. While the $\pi$-acceptor groups lead to planar conformations, the electron-donor groups give rise to non-planar conformations. Isodesmic substituent stabilization energies relative to alkenes have been calculated and correlation with group electronegativities established. Role of induction effect by the substituent groups and resonance effects in charge distribution in the molecules has been analyzed. An analysis of the asymmetric stretching frequencies and intensities of the C=C=O group shows that affect of non-$\pi$ acceptor substituents on the frequency is determined by the field effect (F) and resonance effect (R) parameters, the calculated intensities I (km/mol.) are correlated to group electronegativities $x$ of the substituents by the relationship I = 640.2–100.1 $x$ (r = 0.92). The $\pi$-acceptor substituents increase the intensity which may be explained in terms of their delocalizing effect on the negative charge at the $C_{\beta}$ atom.

• 한국결정학회지
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• 제16권2호
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• pp.102-106
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• 2005

The crystal structure of the title compound has been analyzed by single crystal X-ray diffraction method. The compound crystallized in the triclinic space group $P\bar{1}$ with a=11.300(6) ${\AA}$, c=18.716(10) ${\AA}$, ${\alpha}=82.833(10)^{\circ}$, ${\beta}=83.042(11)^{\circ}$, ${\gamma}=66.139(10)^{\circ}$, $V=2162(2)\;{\AA}^{3}$, Z=2 and R1=0.661 for 10578 unique reflections. The four $C_{5}Me_{5}$ planar groups from a tetrahedron with a mean dihedral angle $70.92(9)^{\circ}$ among them and the $Ti_{4}O_{6}$ cage sits at the center of the tetrahedron. Each Ti atom in the $Ti_{4}O_{6}$ cage is bonded by three bridging oxygen atoms and coordinated by a $C_{5}Me_{5}$ ligand with a mean distance $2.067{\AA}$ from Ti atoms to the centroids of the four five-membered rings. Two oxygen atoms facing each other in $Ti_{4}O_{6}$ cage are $4.051(3){\AA}$ away in average.

• Bulletin of the Korean Chemical Society
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• 제34권12호
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• pp.3771-3776
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• 2013

1-Alkyl-3-methylimidazolium chloride-LiCl melts were prepared from the reaction of 1-alkyl-3-methylimidazolium chloride ([RMIm]Cl; R=allyl or n-butyl) and lithium chloride, and their ability to dissolve cellulose was evaluated. The solubility of cellulose was greatly increased to 320% when [RMIm]Cl was replaced by [RMIm][$LiCl_2$]. Dissolved cellulose in LiCl/[RMIm]Cl melts was successfully regenerated by adding water and LiCl/[RMIm]Cl melts were easily recovered by removing water. As supported by the computational results, the higher solubility of cellulose in [RMIm][$LiCl_2$] can be ascribed to the increased bond distance between anion and C(2)-H of the imidazolium ring compared with that in [RMIm]Cl, thereby resulting in the increased interaction between $[LiCl_2]^-$ and the hydroxyl groups of cellulose.