M-群与内超可解群

M-groups and inner supersolvable groups

M 群的一个著名的结果是:超可解群是M 群.沿着这一方向已有结果:可解外超可解群是M 群;极小非超可解群是M 群.证明了:内超可解群不一定是M 群,即验证了四次交错群A4是内超可解的M 群,而特殊线性群SL(2,3)是内超可解的非M 群.而且给出了内超可解群是M 群的一个充分条件:若G是内超可解群,Φ(G)是G的Frattini子群(即G的所有极大子群的交),那么G/Φ(G)是M 群.注意到,在这一假设下,G/Φ(G)也是一个内超可解群.

It is wellknown that supersolvable groups are Mgroups and it is also known that minimal nonsupersolvable groups are Mgroups,and solvable outer supersolvable groups are Mgroups. In this paper, it is proven that the inner supersolvable groups need not be Mgroups. Also, it is shown that if G is inner supersolvable, then G/Φ(G) is an Mgroup,where Φ(G) denotes the Frattini subgroup of G.