具有较少非循环子群共轭类的有限群

Finite groups with few conjugacy classes of non-cyclic subgroups

设G是有限群,用δ(G)表示群G的非循环子群的共轭类数,πr(G)表示整除|G|的素因子的集合.本文主要研究满足条件δ(G)≤|π(G)|+1的有限群,得到这类群可解,并给出它们的同构分类进一步证明,δ(G)=|π(G)|+2的有限非可解群必同构于A_5或SL(2,5).

Let G be a finite group and δ（G） denote the number of conjugacy classes of all non-cyclic subgroups of G.The symbol π（G） denotes the set of the prime divisors of ｜G｜.In this paper,a group G is called δπ-group if δ（G）≤｜π（G）｜ ＋ 1.We show all δπ-groups are solvable.Furthermore,δπ-groups are classified completely and non-solvable groups with δ（G） = ｜π（G）｜ ＋ 2 must be isomorphic to A5 or SL（2,5）.