| In this paper, we give the sub-cover-avoidance property in a finite group. Let G be a finite group and H a subgroup of G . H is said to have the sub-cover-avoidance property in G if there is a chief series 1=G0＜G1＜…＜Gl=G of G such that for every i=1,…,l, Gi-1(H∩Gi)(?)G . In this paper, we give some necessary and sufficient conditions of a finite group to be solvable under the assumptions that some subgroups of a finite group satisfies the sub-cover-avoidance property. The main results are as follows:Theorem 3.3 Let G be a finite group. Then the following statements are equivalent:(1) G is solvable;(2) Every subgroup of G has the sub-cover-avoiding in G ;(3) Every maximal subgroup of G has the sub-cover-avoiding in G ;(4) Every Hall subgroup of G has the sub-cover-avoiding in G ;(5) Every Sylow subgroup of G has the sub-cover-avoiding in G ;(6) There is a Sylow subgroup P of G for every p∈π(G) such that P has the sub-cover-avoidance property in G ;(7) There is a Sylow subgroup P of G for every p∈π(G) such that P has the p -sub-cover-avoidance property in G .Theorem 3.5 Let G be a finite group and p∈π(G). If p≤minπ(Soc(G)) or(p-1,|G|)=1, then the following statements are equivalent:(1) G is p -solvable;(2) Every maximal subgroup of each Sylow subgroup of G has the sub-cover-avoidance property in G ;(3) There is a Sylow subgroup P of G such that every maximal subgroup of P has the sub-cover-avoidance property in G ;(4) There is a Sylow subgroup P of G such that P contains a maximal subgroup having the p -sub-cover-avoidance property in G .
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