【作者】 余海洋；

【导师】 魏贵民； 曹金文；

【作者基本信息】 成都理工大学 ， 应用数学， 2007， 硕士

【摘要】 有限群论是群论的基础部分，可解群是群论中一类比较常见的群，也是一类极其重要的群。许多群论专家已经得到诸多关于有限群可解的充分条件，有许多结论是研究有限群结构时有用的工具。本文的出发点就是在这些结论的基础上结合Sylow子群、Hall子群、共轭置换子群、c-正规子群等对有限群的可解性进行研究，得到以下主要结论：（1）若G的Sylow 2-子群为交换群，且对G的任意Sylow 2-子群Q（Q≠P），P∩Q在P中极大，则G为可解群。（2）设G是偶阶群，P∈Syl₂（G），若P在G中c-正规，则G为可解群。（3）设M是G的极大子群而且是幂零群，如果M的Sylow 2-子群在G中是c-正规的，则G为可解群。（4）设G是有限群，H是G的偶阶幂零Hall子群，M是H的极大子群，若M的Sylow 2-子群在G中是c-正规，则G是可解群。（5）设H是G的偶阶π-Hall子群，若H及H的每个Sylow子群均在G中共轭置换，则G可解。（6）设P为有限群G的Sylow P-子群，若P在G中共轭置换且G/P的极大子群为1，则G为可解群。（7）设H是G的偶阶π-Hall子群，且H的每个Sylow子群都是正规的，又H在G中共轭置换，则G可解。（8）设H是G的π-Hall子群，且2∈π，H幂零且在G中共轭置换，则G可解。更多还原

【Abstract】 Finite group theory is the basic part of group theory. The solvable group is a kindof ordinary and important group. Many group experts have got some sufficiency aboutthe solvability, many of which are useful tools in researching the structure of finitegroups. In this paper, the starting point is researching the solvability, in the base ofthe conclusions,Combining Sylow-subgroups, Hall-subgroups, conjugate-permutablesubgroups and c-normal groups. We get the following conclusions:（1） If G is sylow2-subgroups are Abelian, and P∩Q is enormous in P forany Sylow 2-subgroups Q（Q≠P）,then G is solvable group.（2） Let G be a group of even order, P∈Syl₂（G）, if P is C-normal inG, then G is solvable group.（3） Let M be a maximal subgroup and nilpotent group of G, if the Sylow2-subgroup of M in G is c-normal, then G is solvable group.（4） Let G be a finite group, H is nilpotent Hall-subgroups with even order ofG. M’s Sylow 2-subgroup is c-normal in G, then G is solvable group.（5） Let H be aπ-Hall subgroup with even order of G, if H and its everySylow-subgroup are all conjugate-permutable, then G is solvable.（6） Let P be a Sylow P-subgroup of G, If P is conjugate-permutable inG and G/P’s maximal subgroup is 1, then G is solvable.（7） Let H be aπ-Hall subgroup with even order of group G, if H’everySylow-subgroup of H is conjugate-permutable in G, then G is solvable.（8） Let H beaπ-Hall subgroup of G, and 2∈π, if H is nilpotent andconjugate-permutable in G, then G is solvable.更多还原