具有小秩Sylow-子群的有限可解群

On finite solvable groups with Sylow-subgroups of small rank

设G是有限p-群,■,其中E为G的初等交换子群,称为G的秩。再令■,其中E为G的初等交换子群,称为G的正规秩。研究Sylow-子群的正规秩≤3的可解群的结构问题,运用极小阶反例法,证明了若G为有限可解群且G的Sylow-子群的正规秩≤3,则G∈N_(2′)N_(2′)N_2U。更进一步,群G的幂零长不超过5,且对所有的素数p,G的p-长不超过2。

We assume G is a finite p-group, r(G)=max{logp|E|,E≤G}.Wherein, E is an elementary exchange subgroup of G, called rank of G. We make rn(G)=max{logp|E|,E≤G,E?G}, where E is an elementary called normal rank of G. The structure problem of solvable groups of normal rank of Sylow-subgroups ≤3 was studied. The method of minimal counterexample was used to prove If G is a finite soluble group and the normal rank of Sylow-subgroup of G≤3, then G∈N2′N2′N2U. Moreover, the nilpotent length of G is no more than 5 and for every prime numbe p, the p-length of G is no more than 2.