两种由群的元素性质所确定的群的根性

讨论群的元素性质与群的根性之间的关系．考虑群的元素性质汐满足条件（1）：如果x是p-的p-元素，且N G，那么xN是G／N的p-元素；满足条件（4）：设N G，如果x∈N是G的p-元素，那么x也是N的p-元素，则群类P1={G}G的每个同态像没有非平凡的p-元素}是p1-根群类．如果p满足条件（1）和条件（2），x∈N是N的p-元素，且N G，那么x也是G的p-元素，则群类P2={G｜G的每个非平凡的同态像有一个非平凡的次正规p-子群}是p-根群类．如果性质p-满足条件（1），（2）和（4），那么P，P2均为根群类且P=P2．

Two Radical Properties of Groups Determined by the Properties of Elements of Groups

In this paper, we make a discussion of the relationship between the properties of the elements of groups and the radical properties of groups. If the properties p of the elements of groups satisfies the conditions （1） if x is p-element of group G and N G, then xN is p-element of group G/N, and （4） let N G, if x 6 N is p-element of group G ,the x is p-element of 5/,then the class of groups P1 = { G ｜ every image of G has not nontrivial p-elements} is p1-class of radical groups. If p satisfies the class of groups P2 = { G ｜ every nontrivial image of G has a nontrivial p- subnormal subgroup } is p2-class of radical groups. If p satisfies the conditions （1）, （2） and （4）, then p, p1 are class of radical of groups, and P = P1.