• Title, Summary, Keyword: fitting subgroup

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  • Shen, Zhencai;Shi, Wujie;Zhang, Jinshan
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.6
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    • pp.1147-1155
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    • 2011
  • In J. Korean Math. Soc, Zhang, Xu and other authors investigated the following problem: what is the structure of finite groups which have many normal subgroups? In this paper, we shall study this question in a more general way. For a finite group G, we define the subgroup $\mathcal{A}(G)$ to be intersection of the normalizers of all non-cyclic subgroups of G. Set $\mathcal{A}_0=1$. Define $\mathcal{A}_{i+1}(G)/\mathcal{A}_i(G)=\mathcal{A}(G/\mathcal{A}_i(G))$ for $i{\geq}1$. By $\mathcal{A}_{\infty}(G)$ denote the terminal term of the ascending series. It is proved that if $G=\mathcal{A}_{\infty}(G)$, then the derived subgroup G' is nilpotent. Furthermore, if all elements of prime order or order 4 of G are in $\mathcal{A}(G)$, then G' is also nilpotent.


  • Kim, Young-Rim;Park, Ju-Mi;Song, Kwang-Yeob
    • The Journal of Korean Academy of Prosthodontics
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    • v.45 no.4
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    • pp.546-554
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    • 2007
  • Statement of problem: Die materials require abrasion resistance, dimensional stability with time, and high surface wettability for adequate material properties. Wear of gypsum materials is a significant problem in the fabrication of accurately fitting cast prosthetic devices. So It has been recommended that the use of die hardener before carving or burnishing of the wax pattern. Purpose: The purpose of this study was to compare the abrasion resistance and surface microhardness(Knoop) with 3 commonly used gypsum die materials(MG Crystal Rock, Super plumstone, GC $FUJIROCK^{(R)}$ EP) with and without the application of 2 die hardeners. Material and methods: Three die materials were evaluated for abrasion resistance and surface microhardness after application of 2 die hardeners(Die hardener and Stone die & plaster hardener). Thirty specimens of each gypsum material were fabricated using an impression of resin die(Pattern resin; GC Corporation, Japan) with 1-mm high ridges, sloped 90 degrees. Gypsum materials were mixed according to manufacturer's recommendations and allowed to set 24 hours before coating. Specimens were arbitrary assigned to 1 of 3 treatment subgroups (n=10/subgroup): no treatment(control), coated with Die hardener, and coated with Stone die & plaster hardener. Abrasion resistance(measured by weight loss) was evaluated using device in 50g mass perpendicular to the ridges. Knoop hardness was determined by loading each specimen face 5 times for 15 seconds with a force of 50g. A scanning electron microscope was used to evaluate the surface of specimens in each treatment subgroup. Conclusions: The obtained results were as follows: 1. 3 types of die stone evaluated in this study did not show significant differences in surface hardness and abrasive resistance(P<.05). 2. In the abrasive resistance test, there were no significant differences between GC $FUJIROCK^{(R)}$ EP and MG Crystal Rock with or without 2 die hardener(P<.05). 3. Super plumstone treated with Stone die & plaster hardener showed increased wear loss(P<.05) 4. Die hardener coatings used in this study decreased the surface hardness of the gypsum material(P<.05).