The structure of a finite simple group and prime graphs

我们研究了有限单群的结构，利用有限单群和素图的分类，得到了有限群的几个性质，Hiroyoshi Yamaki证明了：如果G是偶数阶群，且不是2-秩1，则G中存在一个对合t使得G的阶被G中中心化子t的阶的立方有界.Yutaka Hiramine研究了差和相对非阿贝尔群的差集。特别是Hiramine证明了在5个字母的交错群中恰好存在两个相对差集。此外，Hiramine构造了二面体群中的相对差集，Naoki Chigira，Masaaki Kitazaki和Masaaki Harada构造了自对偶码，使得2^<nd>Janko群充当自同构群。直树Chigira给出了11和12字母Mathieu群的初等构造，饭赖伸夫研究了有限群的素图和可解图，定义了Burn边环。Masato Sawabe证明了由有限群G的所有中心p-根式子群组成的G-偏序集的约化Lefschetz模是X-投射虚Zp[G]-模，其中X是G的非中心p-根式子群的正规化子的p-子群族。

We studied about the structure of finite simple groups and found several properties of finite groups using the classification of finite simple groups and prime graphs.Hiroyoshi Yamaki proved that if G is a group of even order and is not 2-rank 1, then there exists an involution t in G such that the order G is bounded by the cube of the order of the centralizer t in G.Yutaka Hiramine studied about difference and relative difference sets for non- abelian groups. Especially Hiramine proved that there exists exactly two relative difference sets in the alternating group on 5 letters. Further more Hiramine constructed relative difference sets in dihedral groups.Naoki Chigira, Masaaki Kitazume and Masaaki Harada constructed self dual code such that the 2^<nd> Janko group acts as automorphism group. Naoki Chigira gave an elementary construction of the Mathieu groups on 11 and 12 letters.Nobuo Iiyori studied prime graphs and solvable graphs of finite groups and defined Burn side ring. Using these it has been proved that we can describe on the connectedness of the solvable graph by the properties of ideals on Burnside ring.Masato Sawabe proved that the reduced Lefschetz module of the G-poset consisting of all centric p-radical subgroups of a finite group G is X-projective virtual Zp[G]-module where X is a family of p-subgroups of the normalizers of non-centric p-radical subgroups of G.