The Research On The Solvability Of Finite Group

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 aÏ€-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 aÏ€-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 beaÏ€-Hall subgroup of G, and 2∈π, if H is nilpotent andconjugate-permutable in G, then G is solvable.