关键词: 有限群, 可解群, 极大子群, C－正规子群

Abstract: It is proved in this paper‚if some subgroups of the finite group G‚such as the maximal subgroup‚Sylow subgroup and its subgroups‚are also C-normal subgroups of group G‚some sufficient conditions as well as necessary and sufficient conditions for the solvability of group G can be obtained．By determining whether its subgroups are Cnormal subgroups‚the solvability of group G can be determined．In the proof‚the method of contradiction of the extreme minimal rank of group G is adopted．This method is a combination of induction and reduction to absurdity．Besides‚a new subgruop set L（G）‚that is‚the maximal subgroups excluding the guided group of G is introduced into the proof． 

Key words: finite group, solvable group, maximal subgroup, C-normal subgroup