有限P-可解群的P-长

On the p-length of finite p-soluble groups

如果有限群G有次正规列,即1=G0?G1?…?Gt=G,对任意的i∈{1,2,…,t},都有截面Gi/G(i-1)或为交换群或为p′-群,则群G被称为p-可解群。通过对特殊p-群、超特殊p-群的性质分析,讨论了饱和群系中p-长不等于1的p-可解群的一些性质。应用极小阶反证法证明:若■是一个饱和群系,并且群G是一个p-长不等于1的p-可解群,用非空集S(G)表示G所有截面A/B的集合,如果满足截面A/B的p-长不等于1且截面A/B的一个Sylow p-子群同构于极小非■群K的■-上根,若■或■,那么p=3且S(G)中具有极小阶的所有截面同构于[Z3×Z3]SL2(3)。

A finite group G is called a p-solvable group,if there is a subnormal series 1=G0?G1?…?Gt,and for each i∈{1,2,…,t},Gi/Gi-1 is either abelian group or p’-group.Through analyzing special p-group and extraspecial p-group,some properties of p-solvable groups with p-length unequal to 1 in a saturated formation are discussed.The method of minimal-order counter example is used to prove that:if■is a saturated formation,and G is a p-soluble group with p-length≠1,S(G)≠?represents the set of all sections A/B of G.If p-length of A/B section is unequal to 1,and a Sylow p-subgroup of A/B is isomorphic to the■-co-radical of minimal non-■-group K;if■or■,then p=3 and any section of minimal order in S(G)is isomorphic to[Z3×Z3]SL2(3).