关于极大子群共轭类型的若干结果

Some results on the type of conjugacy classes of maximal subgroups

设G为恰有k个极大子群共轭类的有限群,m1,…,mk分别为相应的共轭类长且m1≤…≤mk,则称MT(G)=(m1,…,mk)为有限群G的极大子群共轭类型。主要研究问题:设K为一个群类,G和N为有限群。若MT(G)=MT(N),是否恒有G属于K当且仅当N属于K?特别地,讨论G/Φ(G)和N/Φ(N)之间的关系。

Let G be a finite group with k conjugacy classes of maximal subgroups,and m1 ,...,mk be the lengthes of this classes and m1 ≤…≤mk. Then the sequence MT（G） = （m1,..., mk ）is called the type of conjugacy classes of maximal subgroups of G. In this artcle, the following problems are maily investigated. Let .H be a class of groups, G and N be finite groups. If MT（G） = MT（ N）, can we ensure that G is in H if and only if N is in H？ In particular,what can we say about G/Ф（G） and N/Ф （N）.