三维晶体学群的极大有限群的代数基础

从 NTN =T这一晶体学的普遍公式出发 ,推导出在三维空间中度量张量矩阵T有四个算术不等价类 ,即T1 =10 00 100 0 1,T2 =2 1012 00 0 2,T3=2 1112 1112,T4=3- 1- 1- 13- 1- 1- 13,而T1 ,T3,T4属几何等价类 ,故几何不等价类只有T1 及T2 .根据 NT1 N =T1 及 NT2 N =T2 求出三维晶体学的两个极大有限群分别为 48阶及 2 4阶 ,它们对应于两个晶体学点群 。

Finite groups whose maximal subgroups of sylow p-subgroups admit a p-solvable supplement

In this note,we show that if every maximal subgroup of a Sylow p-subgroup of a finite group has a p-solvable supplement then the group is necessarily p-solvable.This gives a positive answer to Problem 17.111 of the Kourovka Notebook(Unsolved Problems in Group Theory),which was posed by Skiba.

Finite groups with systems of Σ-embedded subgroups

In recent years,a series of papers about cover-avoiding property of subgroups appeared and all the studies were connected with chief factors of a finite group.However,about the cover-avoiding property of subgroups for non-chief factor,there is no study up to now.The purpose of this paper is to build the theory.Let A be a subgroup of a finite group G and Σ:G0≤G1≤…≤Gn some subgroup series of G.Suppose that for each pair(K,H) such that K is a maximal subgroup of H and G i 1 K < H G i for some i,either A ∩ H = A ∩ K or AH = AK.Then we say that A is Σ-embedded in G.In this paper,we study the finite groups with given systems of Σ-embedded subgroups.The basic properties of Σ-embedded subgroups are established and some new characterizations of some classes of finite groups are given and some known results are generalized.

Finite p-Groups all of Whose Maximal Subgroups Either are Metacyclic or Have a Derived Subgroup of Order ≤ p

The groups as mentioned in the title are classified up to isomorphism. This is an answer to a question proposed by Berkovich and Janko.

Finite Groups All of Whose Second Maximal Subgroups Are PSC*-Groups

This paper discusses the influence of minimal subgroups on the structure of finite groups and gives the structures of finite groups all of whose second maximal subgroups are PSC*-groups.